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<item>
<title>Erratum for “An inverse theorem for the Gowers U^{s+1}[N]-norm”</title>
<link>https://terrytao.wordpress.com/2024/04/25/erratum-for-an-inverse-theorem-for-the-gowers-us1n-norm/</link>
<comments>https://terrytao.wordpress.com/2024/04/25/erratum-for-an-inverse-theorem-for-the-gowers-us1n-norm/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Thu, 25 Apr 2024 15:41:46 +0000</pubDate>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[update]]></category>
<category><![CDATA[Ben Green]]></category>
<category><![CDATA[erratum]]></category>
<category><![CDATA[Gowers uniformity norms]]></category>
<category><![CDATA[Tamar Ziegler]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14486</guid>
<description><![CDATA[The purpose of this post is to report an erratum to the 2012 paper “An inverse theorem for the Gowers -norm” of Ben Green, myself, and Tamar Ziegler (previously discussed in this blog post). The main results of this paper have been superseded with stronger quantitative results, first in work of Manners (using somewhat different […]]]></description>
<content:encoded><![CDATA[

The purpose of this post is to report an <a href="https://terrytao.files.wordpress.com/2024/04/erratum-4.pdf">erratum</a> to the 2012 paper “<a href="https://zbmath.org/1282.11007">An inverse theorem for the Gowers <img src="https://s0.wp.com/latex.php?latex=%7BU%5E%7Bs%2B1%7D%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%5E%7Bs%2B1%7D%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%5E%7Bs%2B1%7D%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U^{s+1}[N]}" class="latex" />-norm</a>” of <a href="https://people.maths.ox.ac.uk/greenbj/">Ben Green</a>, myself, and <a href="https://ma.huji.ac.il/~tamarz/">Tamar Ziegler</a> (previously discussed in <a href="https://terrytao.wordpress.com/2010/09/21/an-inverse-theorem-for-the-gowers-us1n-norm/">this blog post</a>). The main results of this paper have been superseded with stronger quantitative results, first in <a href="https://arxiv.org/abs/1811.00718">work of Manners</a> (using somewhat different methods), and more recently in a remarkable <a href="https://arxiv.org/abs/2402.17994">paper of Leng, Sah, and Sawhney</a> which combined the methods of our paper with several new innovations to obtain quite strong bounds (of quasipolynomial type); see also an alternate proof of our main results (again by quite different methods) <a href="https://zbmath.org/1503.37012">by Candela and Szegedy</a>. In the course of their work, they discovered some fixable but nontrivial errors in our paper. These (rather technical) issues were already implicitly corrected in this followup work which supersedes our own paper, but for the sake of completeness we are also providing a formal erratum for our original paper, which can be found <a href="https://terrytao.files.wordpress.com/2024/04/erratum-4.pdf">here</a>. We thank Leng, Sah, and Sawhney for bringing these issues to our attention.

Excluding some minor (mostly typographical) issues which we also have reported in this erratum, the main issues stemmed from a conflation of two notions of a degree <img src="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bs%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{s}" class="latex" /> filtration <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++G+%3D+G_0+%5Cgeq+G_1+%5Cgeq+%5Cdots+%5Cgeq+G_s+%5Cgeq+G_%7Bs%2B1%7D+%3D+%5C%7B1%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++G+%3D+G_0+%5Cgeq+G_1+%5Cgeq+%5Cdots+%5Cgeq+G_s+%5Cgeq+G_%7Bs%2B1%7D+%3D+%5C%7B1%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++G+%3D+G_0+%5Cgeq+G_1+%5Cgeq+%5Cdots+%5Cgeq+G_s+%5Cgeq+G_%7Bs%2B1%7D+%3D+%5C%7B1%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle G = G_0 \geq G_1 \geq \dots \geq G_s \geq G_{s+1} = \{1\}" class="latex" />
of a group <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, which is a nested sequence of subgroups that obey the relation <img src="https://s0.wp.com/latex.php?latex=%7B%5BG_i%2CG_j%5D+%5Cleq+G_%7Bi%2Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5BG_i%2CG_j%5D+%5Cleq+G_%7Bi%2Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5BG_i%2CG_j%5D+%5Cleq+G_%7Bi%2Bj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{[G_i,G_j] \leq G_{i+j}}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=%7Bi%2Cj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%2Cj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%2Cj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i,j}" class="latex" />. The weaker notion (sometimes known as a prefiltration) permits the group <img src="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_1}" class="latex" /> to be strictly smaller than <img src="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_0}" class="latex" />, while the stronger notion requires <img src="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_0}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_1}" class="latex" /> to equal. In practice, one can often move between the two concepts, as <img src="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_1}" class="latex" /> is always normal in <img src="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_0}" class="latex" />, and a prefiltration behaves like a filtration on every coset of <img src="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_1}" class="latex" /> (after applying a translation and perhaps also a conjugation). However, we did not clarify this issue sufficiently in the paper, and there are some places in the text where results that were only proven for filtrations were applied for prefiltrations. The erratum fixes this issues, mostly by clarifying that we work with filtrations throughout (which requires some decomposition into cosets in places where prefiltrations are generated). Similar adjustments need to be made for multidegree filtrations and degree-rank filtrations, which we also use heavily on our paper.

In most cases, fixing this issue only required minor changes to the text, but there is one place (Section 8) where there was a non-trivial problem: we used the claim that the final group <img src="https://s0.wp.com/latex.php?latex=%7BG_s%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG_s%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG_s%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G_s}" class="latex" /> was a central group, which is true for filtrations, but not necessarily for prefiltrations. This fact (or more precisely, a multidegree variant of it) was used to claim a factorization for a certain product of nilcharacters, which is in fact not true as stated. In the erratum, a substitute factorization for a slightly different product of nilcharacters is provided, which is still sufficient to conclude the main result of this part of the paper (namely, a statistical linearization of a certain family of nilcharacters in the shift parameter <img src="https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h}" class="latex" />).

Again, we stress that these issues do not impact the paper of Leng, Sah, and Sawhney, as they adapted the methods in our paper in a fashion that avoids these errors.

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<slash:comments>5</slash:comments>
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<media:title type="html">Terry</media:title>
</media:content>
</item>
<item>
<title>Notes on the B+B+t theorem</title>
<link>https://terrytao.wordpress.com/2024/04/24/notes-on-the-bbt-theorem/</link>
<comments>https://terrytao.wordpress.com/2024/04/24/notes-on-the-bbt-theorem/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Wed, 24 Apr 2024 15:02:49 +0000</pubDate>
<category><![CDATA[expository]]></category>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[math.DS]]></category>
<category><![CDATA[Bryna Kra]]></category>
<category><![CDATA[Donald Robertson]]></category>
<category><![CDATA[Ethan Acklesberg]]></category>
<category><![CDATA[Florian Richter]]></category>
<category><![CDATA[infinitary mathematics]]></category>
<category><![CDATA[Joel Moreira]]></category>
<category><![CDATA[sumsets]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14475</guid>
<description><![CDATA[A recent paper of Kra, Moreira, Richter, and Robertson established the following theorem, resolving a question of Erdös. Given a discrete amenable group , and a subset of , we define the Banach density of to be the quantity where the supremum is over all Følner sequences of . Given a set in , we […]]]></description>
<content:encoded><![CDATA[

A <a href="https://arxiv.org/abs/2206.12377">recent paper of Kra, Moreira, Richter, and Robertson</a> established the following theorem, resolving a <a href="https://zbmath.org/0305.10050">question of Erdös</a>. Given a discrete amenable group <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G = (G,+)}" class="latex" />, and a subset <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, we define the <a href="https://en.wikipedia.org/wiki/Natural_density">Banach density</a> of <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> to be the quantity <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_%5CPhi+%5Climsup_%7BN+%5Crightarrow+%5Cinfty%7D+%7CA+%5Ccap+%5CPhi_N%7C%2F%7C%5CPhi_N%7C%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_%5CPhi+%5Climsup_%7BN+%5Crightarrow+%5Cinfty%7D+%7CA+%5Ccap+%5CPhi_N%7C%2F%7C%5CPhi_N%7C%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_%5CPhi+%5Climsup_%7BN+%5Crightarrow+%5Cinfty%7D+%7CA+%5Ccap+%5CPhi_N%7C%2F%7C%5CPhi_N%7C%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sup_\Phi \limsup_{N \rightarrow \infty} |A \cap \Phi_N|/|\Phi_N|," class="latex" />
where the supremum is over all <a href="https://en.wikipedia.org/wiki/F%C3%B8lner_sequence">Følner sequences</a> <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi+%3D+%28%5CPhi_N%29_%7BN%3D1%7D%5E%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi+%3D+%28%5CPhi_N%29_%7BN%3D1%7D%5E%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi+%3D+%28%5CPhi_N%29_%7BN%3D1%7D%5E%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi = (\Phi_N)_{N=1}^\infty}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />. Given a set <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, we define the restricted sumset <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \oplus B}" class="latex" /> to be the set of all pairs <img src="https://s0.wp.com/latex.php?latex=%7Bb_1%2Bb_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_1%2Bb_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_1%2Bb_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_1+b_2}" class="latex" /> where <img src="https://s0.wp.com/latex.php?latex=%7Bb_1%2C+b_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_1%2C+b_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_1%2C+b_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_1, b_2}" class="latex" /> are distinct elements of <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" />.

<blockquote>Theorem 1 <a name="main"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> be a countably infinite abelian group with the index <img src="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{[G:2G]}" class="latex" /> finite. Let <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> be a positive Banach density subset of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />. Then there exists an infinite set <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \subset A}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t \in G}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \oplus B + t \subset A}" class="latex" />. </blockquote>


Strictly speaking, the main result of Kra et al. only claims this theorem for the case of the integers <img src="https://s0.wp.com/latex.php?latex=%7BG%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G={\bf Z}}" class="latex" />, but as noted in the recent preprint of <a href="https://arxiv.org/abs/2402.07779">Charamaras and Mountakis</a>, the argument in fact applies for all countable abelian <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> in which the subgroup <img src="https://s0.wp.com/latex.php?latex=%7B2G+%3A%3D+%5C%7B+2x%3A+x+%5Cin+G+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2G+%3A%3D+%5C%7B+2x%3A+x+%5Cin+G+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2G+%3A%3D+%5C%7B+2x%3A+x+%5Cin+G+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2G := \{ 2x: x \in G \}}" class="latex" /> has finite index. This condition is in fact necessary (as observed by forthcoming work of Ethan Acklesberg): if <img src="https://s0.wp.com/latex.php?latex=%7B2G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2G}" class="latex" /> has infinite index, then one can find a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H_j}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> of index <img src="https://s0.wp.com/latex.php?latex=%7B2%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^j}" class="latex" /> for any <img src="https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j \geq 1}" class="latex" /> that contains <img src="https://s0.wp.com/latex.php?latex=%7B2G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2G}" class="latex" /> (or equivalently, <img src="https://s0.wp.com/latex.php?latex=%7BG%2FH_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%2FH_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%2FH_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G/H_j}" class="latex" /> is <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />-torsion). If one lets <img src="https://s0.wp.com/latex.php?latex=%7By_1%2Cy_2%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By_1%2Cy_2%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By_1%2Cy_2%2C%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y_1,y_2,\dots}" class="latex" /> be an enumeration of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, and one can then check that the set <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3A%3D+G+%5Cbackslash+%5Cbigcup_%7Bj%3D1%7D%5E%5Cinfty+%28H_%7Bj%2B1%7D+%2B+y_j%29+%5Cbackslash+%5C%7By_1%2C%5Cdots%2Cy_j%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3A%3D+G+%5Cbackslash+%5Cbigcup_%7Bj%3D1%7D%5E%5Cinfty+%28H_%7Bj%2B1%7D+%2B+y_j%29+%5Cbackslash+%5C%7By_1%2C%5Cdots%2Cy_j%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++A+%3A%3D+G+%5Cbackslash+%5Cbigcup_%7Bj%3D1%7D%5E%5Cinfty+%28H_%7Bj%2B1%7D+%2B+y_j%29+%5Cbackslash+%5C%7By_1%2C%5Cdots%2Cy_j%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle A := G \backslash \bigcup_{j=1}^\infty (H_{j+1} + y_j) \backslash \{y_1,\dots,y_j\}" class="latex" />
has positive Banach density, but does not contain any set of the form <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \oplus B + t}" class="latex" /> for any <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> (indeed, from the pigeonhole principle and the <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />-torsion nature of <img src="https://s0.wp.com/latex.php?latex=%7BG%2FH_%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%2FH_%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%2FH_%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G/H_{j+1}}" class="latex" /> one can show that <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+y_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+y_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+y_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \oplus B + y_j}" class="latex" /> must intersect <img src="https://s0.wp.com/latex.php?latex=%7BH_%7Bj%2B1%7D+%2B+y_j+%5Cbackslash+%5C%7By_1%2C%5Cdots%2Cy_j%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH_%7Bj%2B1%7D+%2B+y_j+%5Cbackslash+%5C%7By_1%2C%5Cdots%2Cy_j%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH_%7Bj%2B1%7D+%2B+y_j+%5Cbackslash+%5C%7By_1%2C%5Cdots%2Cy_j%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H_{j+1} + y_j \backslash \{y_1,\dots,y_j\}}" class="latex" /> whenever <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> has cardinality larger than <img src="https://s0.wp.com/latex.php?latex=%7Bj+2%5E%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj+2%5E%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj+2%5E%7Bj%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j 2^{j+1}}" class="latex" />). It is also necessary to work with restricted sums <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \oplus B}" class="latex" /> rather than full sums <img src="https://s0.wp.com/latex.php?latex=%7BB%2BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%2BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%2BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B+B}" class="latex" />: a counterexample to the latter is provided for instance by the example with <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G = {\bf Z}}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BA+%3A%3D+%5Cbigcup_%7Bj%3D1%7D%5E%5Cinfty+%5B10%5Ej%2C+1.1+%5Ctimes+10%5Ej%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%3A%3D+%5Cbigcup_%7Bj%3D1%7D%5E%5Cinfty+%5B10%5Ej%2C+1.1+%5Ctimes+10%5Ej%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%3A%3D+%5Cbigcup_%7Bj%3D1%7D%5E%5Cinfty+%5B10%5Ej%2C+1.1+%5Ctimes+10%5Ej%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A := \bigcup_{j=1}^\infty [10^j, 1.1 \times 10^j]}" class="latex" />. Finally, the presence of the shift <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> is also necessary, as can be seen by considering the example of <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> being the odd numbers in <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G ={\bf Z}}" class="latex" />, though in the case <img src="https://s0.wp.com/latex.php?latex=%7BG%3D2G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%3D2G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%3D2G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G=2G}" class="latex" /> one can of course delete the shift <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> at the cost of giving up the containment <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \subset A}" class="latex" />.

Theorem <a href="#main">1</a> resembles other theorems in density Ramsey theory, such as <a href="https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem">Szemerédi’s theorem</a>, but with the notable difference that the pattern located in the dense set <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is infinite rather than merely arbitrarily large but finite. As such, it does not seem that this theorem can be proven by purely finitary means. However, one can view this result as the conjunction of an infinite number of statements, each of which is a finitary density Ramsey theory statement. To see this, we need some more notation. Observe from Tychonoff’s theorem that the collection <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG+%3A%3D+%5C%7B+B%3A+B+%5Csubset+G+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG+%3A%3D+%5C%7B+B%3A+B+%5Csubset+G+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG+%3A%3D+%5C%7B+B%3A+B+%5Csubset+G+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G := \{ B: B \subset G \}}" class="latex" /> is a compact topological space (with the topology of pointwise convergence) (it is also metrizable since <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> is countable). Subsets <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+F%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+F%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+F%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal F}}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G}" class="latex" /> can be thought of as properties of subsets of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />; for instance, the property a subset <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> of being finite is of this form, as is the complementary property of being infinite. A property of subsets of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> can then be said to be closed or open if it corresponds to a closed or open subset of <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G}" class="latex" />. Thus, a property is closed and only if if it is closed under pointwise limits, and a property is open if, whenever a set <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> has this property, then any other set <img src="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B'}" class="latex" /> that shares a sufficiently large (but finite) initial segment with <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> will also have this property. Since <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G}" class="latex" /> is compact and Hausdorff, a property is closed if and only if it is compact.

The properties of being finite or infinite are neither closed nor open. Define a smallness property to be a closed (or compact) property of subsets of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> that is only satisfied by finite sets; the complement to this is a largeness property, which is an open property of subsets of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> that is satisfied by all infinite sets. (One could also choose to impose other axioms on these properties, for instance requiring a largeness property to be an <a href="https://en.wikipedia.org/wiki/Upper_set">upper set</a>, but we will not do so here.) Examples of largeness properties for a subset <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> include:
<ul> <li> <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> has at least <img src="https://s0.wp.com/latex.php?latex=%7B10%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B10%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B10%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{10}" class="latex" /> elements. </li><li> <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> is non-empty and has at least <img src="https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_1}" class="latex" /> elements, where <img src="https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_1}" class="latex" /> is the smallest element of <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" />. </li><li> <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> is non-empty and has at least <img src="https://s0.wp.com/latex.php?latex=%7Bb_%7Bb_1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_%7Bb_1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_%7Bb_1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_{b_1}}" class="latex" /> elements, where <img src="https://s0.wp.com/latex.php?latex=%7Bb_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_n}" class="latex" /> is the <img src="https://s0.wp.com/latex.php?latex=%7Bn%5E%7B%5Cmathrm%7Bth%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%5E%7B%5Cmathrm%7Bth%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%5E%7B%5Cmathrm%7Bth%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n^{\mathrm{th}}}" class="latex" /> element of <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" />. </li><li> <img src="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T}" class="latex" /> halts when given <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> as input, where <img src="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T}" class="latex" /> is a given Turing machine that halts whenever given an infinite set as input. (Note that this encompasses the preceding three examples as special cases, by selecting <img src="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T}" class="latex" /> appropriately.)
</li></ul>
We will call a set obeying a largeness property <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}}" class="latex" /> an <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}}" class="latex" />-large set.

Theorem <a href="#main">1</a> is then equivalent to the following “almost finitary” version (cf. <a href="https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/">this previous discussion</a> of almost finitary versions of the infinite pigeonhole principle):

<blockquote>Theorem 2 (Almost finitary form of main theorem) <a name="thm-main-alt"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> be a countably infinite abelian group with <img src="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{[G:2G]}" class="latex" /> finite. Let <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_n}" class="latex" /> be a Følner sequence in <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, let <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdelta%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\delta>0}" class="latex" />, and let <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}_t}" class="latex" /> be a largeness property for each <img src="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t \in G}" class="latex" />. Then there exists <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> such that if <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \subset G}" class="latex" /> is such that <img src="https://s0.wp.com/latex.php?latex=%7B%7CA+%5Ccap+%5CPhi_n%7C+%2F+%7C%5CPhi_n%7C+%5Cgeq+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA+%5Ccap+%5CPhi_n%7C+%2F+%7C%5CPhi_n%7C+%5Cgeq+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA+%5Ccap+%5CPhi_n%7C+%2F+%7C%5CPhi_n%7C+%5Cgeq+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A \cap \Phi_n| / |\Phi_n| \geq \delta}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \leq N}" class="latex" />, then there exists a shift <img src="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t \in G}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> contains a <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}_t}" class="latex" />-large set <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \oplus B + t \subset A}" class="latex" />. </blockquote>


Proof of Theorem <a href="#thm-main-alt">2</a> assuming Theorem <a href="#main">1</a>. Let <img src="https://s0.wp.com/latex.php?latex=%7BG%2C+%5CPhi_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%2C+%5CPhi_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%2C+%5CPhi_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G, \Phi_n}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\delta}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}_t}" class="latex" /> be as in Theorem <a href="#thm-main-alt">2</a>. Suppose for contradiction that Theorem <a href="#thm-main-alt">2</a> failed, then for each <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> we can find <img src="https://s0.wp.com/latex.php?latex=%7BA_N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_N}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7B%7CA_N+%5Ccap+%5CPhi_n%7C+%2F+%7C%5CPhi_n%7C+%5Cgeq+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA_N+%5Ccap+%5CPhi_n%7C+%2F+%7C%5CPhi_n%7C+%5Cgeq+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA_N+%5Ccap+%5CPhi_n%7C+%2F+%7C%5CPhi_n%7C+%5Cgeq+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A_N \cap \Phi_n| / |\Phi_n| \geq \delta}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cleq+N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \leq N}" class="latex" />, such that there is no <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}_t}" class="latex" />-large <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A_N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A_N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A_N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B, B \oplus B + t \subset A_N}" class="latex" />. By compactness, a subsequence of the <img src="https://s0.wp.com/latex.php?latex=%7BA_N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_N}" class="latex" /> converges pointwise to a set <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" />, which then has Banach density at least <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\delta}" class="latex" />. By Theorem <a href="#main">1</a>, there is an infinite set <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> and a <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B, B \oplus B + t \subset A}" class="latex" />. By openness, we conclude that there exists a finite <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}_t}" class="latex" />-large set <img src="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B'}" class="latex" /> contained in <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" />, thus <img src="https://s0.wp.com/latex.php?latex=%7BB%27%2C+B%27+%5Coplus+B%27+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%27%2C+B%27+%5Coplus+B%27+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%27%2C+B%27+%5Coplus+B%27+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B', B' \oplus B' + t \subset A}" class="latex" />. This implies that <img src="https://s0.wp.com/latex.php?latex=%7BB%27%2C+B%27+%5Coplus+B%27+%2B+t+%5Csubset+A_N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%27%2C+B%27+%5Coplus+B%27+%2B+t+%5Csubset+A_N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%27%2C+B%27+%5Coplus+B%27+%2B+t+%5Csubset+A_N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B', B' \oplus B' + t \subset A_N}" class="latex" /> for infinitely many <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" />, a contradiction.

Proof of Theorem <a href="#main">1</a> assuming Theorem <a href="#thm-main-alt">2</a>. Let <img src="https://s0.wp.com/latex.php?latex=%7BG%2C+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%2C+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%2C+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G, A}" class="latex" /> be as in Theorem <a href="#main">1</a>. If the claim failed, then for each <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" />, the property <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cmathcal+P%7D_t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\mathcal P}_t}" class="latex" /> of being a set <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> for which <img src="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B, B \oplus B + t \subset A}" class="latex" /> would be a smallness property. By Theorem <a href="#thm-main-alt">2</a>, we see that there is a <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> and a <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> obeying the complement of this property such that <img src="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%2C+B+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B, B \oplus B + t \subset A}" class="latex" />, a contradiction.

<blockquote>Remark 3 Define a relation <img src="https://s0.wp.com/latex.php?latex=%7BR%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R}" class="latex" /> between <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG+%5Ctimes+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG+%5Ctimes+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG+%5Ctimes+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G \times G}" class="latex" /> by declaring <img src="https://s0.wp.com/latex.php?latex=%7BA%5C+R%5C+%28B%2Ct%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%5C+R%5C+%28B%2Ct%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%5C+R%5C+%28B%2Ct%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A\ R\ (B,t)}" class="latex" /> if <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \subset A}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB+%5Coplus+B+%2B+t+%5Csubset+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B \oplus B + t \subset A}" class="latex" />. The key observation that makes the above equivalences work is that this relation is continuous in the sense that if <img src="https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U}" class="latex" /> is an open subset of <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG+%5Ctimes+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG+%5Ctimes+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG+%5Ctimes+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G \times G}" class="latex" />, then the inverse image <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+R%5E%7B-1%7D+U+%3A%3D+%5C%7B+A+%5Cin+2%5EG%3A+A%5C+R%5C+%28B%2Ct%29+%5Chbox%7B+for+some+%7D+%28B%2Ct%29+%5Cin+U+%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+R%5E%7B-1%7D+U+%3A%3D+%5C%7B+A+%5Cin+2%5EG%3A+A%5C+R%5C+%28B%2Ct%29+%5Chbox%7B+for+some+%7D+%28B%2Ct%29+%5Cin+U+%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+R%5E%7B-1%7D+U+%3A%3D+%5C%7B+A+%5Cin+2%5EG%3A+A%5C+R%5C+%28B%2Ct%29+%5Chbox%7B+for+some+%7D+%28B%2Ct%29+%5Cin+U+%5C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle R^{-1} U := \{ A \in 2^G: A\ R\ (B,t) \hbox{ for some } (B,t) \in U \}" class="latex" />
is also open. Indeed, if <img src="https://s0.wp.com/latex.php?latex=%7BA%5C+R%5C+%28B%2Ct%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%5C+R%5C+%28B%2Ct%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%5C+R%5C+%28B%2Ct%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A\ R\ (B,t)}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7B%28B%2Ct%29+%5Cin+U%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28B%2Ct%29+%5Cin+U%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28B%2Ct%29+%5Cin+U%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(B,t) \in U}" class="latex" />, then <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" /> contains a finite set <img src="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B'}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7B%28B%27%2Ct%29+%5Cin+U%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28B%27%2Ct%29+%5Cin+U%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28B%27%2Ct%29+%5Cin+U%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(B',t) \in U}" class="latex" />, and then any <img src="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A'}" class="latex" /> that contains both <img src="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B'}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BB%27+%5Coplus+B%27+%2B+t%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%27+%5Coplus+B%27+%2B+t%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%27+%5Coplus+B%27+%2B+t%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B' \oplus B' + t}" class="latex" /> lies in <img src="https://s0.wp.com/latex.php?latex=%7BR%5E%7B-1%7D+U%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BR%5E%7B-1%7D+U%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BR%5E%7B-1%7D+U%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{R^{-1} U}" class="latex" />. </blockquote>


For each specific largeness property, such as the examples listed previously, Theorem <a href="#thm-main-alt">2</a> can be viewed as a finitary assertion (at least if the property is “computable” in some sense), but if one quantifies over all largeness properties, then the theorem becomes infinitary. In the spirit of the <a href="https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem">Paris-Harrington theorem</a>, I would in fact expect some cases of Theorem <a href="#thm-main-alt">2</a> to undecidable statements of Peano arithmetic, although I do not have a rigorous proof of this assertion.

Despite the complicated finitary interpretation of this theorem, I was still interested in trying to write the proof of Theorem <a href="#main">1</a> in some sort of “pseudo-finitary” manner, in which one can see analogies with finitary arguments in additive combinatorics. The proof of Theorem <a href="#main">1</a> that I give below the fold is my attempt to achieve this, although to avoid a complete explosion of “<a href="https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/">epsilon management</a>” I will still use at one juncture an ergodic theory reduction from the original paper of Kra et al. that relies on such infinitary tools as the ergodic decomposition, the ergodic theory, and the spectral theorem. Also some of the steps will be a little sketchy, and assume some familiarity with additive combinatorics tools (such as the arithmetic regularity lemma).



 — 1. Proof of theorem — 

The proof of Kra et al. proceeds by establishing the following related statement. Define a (length three) combinatorial Erdös progression to be a triple <img src="https://s0.wp.com/latex.php?latex=%7B%28A%2CX_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28A%2CX_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28A%2CX_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(A,X_1,X_2)}" class="latex" /> of subsets of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> such that there exists a sequence <img src="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_j \rightarrow \infty}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7BA+-+n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+-+n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+-+n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A - n_j}" class="latex" /> converges pointwise to <img src="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX_1-n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1-n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1-n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1-n_j}" class="latex" /> converges pointwise to <img src="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_2}" class="latex" />. (By <img src="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_j \rightarrow \infty}" class="latex" />, we mean with respect to the cocompact filter; that is, that for any finite (or, equivalently, compact) subset <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Cnot+%5Cin+K%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Cnot+%5Cin+K%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_j+%5Cnot+%5Cin+K%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_j \not \in K}" class="latex" /> for all sufficiently large <img src="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{j}" class="latex" />.)

<blockquote>Theorem 4 (Combinatorial Erdös progression) <a name="main-erdos"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> be a countably infinite abelian group with <img src="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{[G:2G]}" class="latex" /> finite. Let <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> be a positive Banach density subset of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />. Then there exists a combinatorial Erdös progression <img src="https://s0.wp.com/latex.php?latex=%7B%28A%2CX_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28A%2CX_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28A%2CX_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(A,X_1,X_2)}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7B0+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 \in X_1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_2}" class="latex" /> non-empty. </blockquote>


Let us see how Theorem <a href="#main-erdos">4</a> implies Theorem <a href="#main">1</a>. Let <img src="https://s0.wp.com/latex.php?latex=%7BG%2C+A%2C+X_1%2C+X_2%2C+n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%2C+A%2C+X_1%2C+X_2%2C+n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%2C+A%2C+X_1%2C+X_2%2C+n_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G, A, X_1, X_2, n_j}" class="latex" /> be as in Theorem <a href="#main-erdos">4</a>. By hypothesis, <img src="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_2}" class="latex" /> contains an element <img src="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, thus <img src="https://s0.wp.com/latex.php?latex=%7B0+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B0+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B0+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{0 \in X_1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+X_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+X_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt+%5Cin+X_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t \in X_2}" class="latex" />. Setting <img src="https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_1}" class="latex" /> to be a sufficiently large element of the sequence <img src="https://s0.wp.com/latex.php?latex=%7Bn_1%2C+n_2%2C+%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_1%2C+n_2%2C+%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_1%2C+n_2%2C+%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_1, n_2, \dots}" class="latex" />, we conclude that <img src="https://s0.wp.com/latex.php?latex=%7Bb_1+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_1+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_1+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_1 \in A}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bb_1+%2B+t+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_1+%2B+t+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_1+%2B+t+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_1 + t \in X_1}" class="latex" />. Setting <img src="https://s0.wp.com/latex.php?latex=%7Bb_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_2}" class="latex" /> to be an even larger element of this sequence, we then have <img src="https://s0.wp.com/latex.php?latex=%7Bb_2%2C+b_2%2Bb_1%2Bt+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_2%2C+b_2%2Bb_1%2Bt+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_2%2C+b_2%2Bb_1%2Bt+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_2, b_2+b_1+t \in A}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bb_2+%2Bt+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_2+%2Bt+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_2+%2Bt+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_2 +t \in X_1}" class="latex" />. Setting <img src="https://s0.wp.com/latex.php?latex=%7Bb_3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_3}" class="latex" /> to be an even larger element, we have <img src="https://s0.wp.com/latex.php?latex=%7Bb_3%2C+b_3%2Bb_1%2Bt%2C+b_3%2Bb_2%2Bt+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_3%2C+b_3%2Bb_1%2Bt%2C+b_3%2Bb_2%2Bt+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_3%2C+b_3%2Bb_1%2Bt%2C+b_3%2Bb_2%2Bt+%5Cin+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_3, b_3+b_1+t, b_3+b_2+t \in A}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bb_3+%2B+t+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bb_3+%2B+t+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bb_3+%2B+t+%5Cin+X_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{b_3 + t \in X_1}" class="latex" />. Continuing in this fashion we obtain the desired infinite set <img src="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B}" class="latex" />.

It remains to establish Theorem <a href="#main-erdos">4</a>. The proof of Kra et al. converts this to a topological dynamics/ergodic theory problem. Define a topological measure-preserving <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />-system <img src="https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X,T,\mu)}" class="latex" /> to be a compact space <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> equipped with a Borel probability measure <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> as well as a measure-preserving homeomorphism <img src="https://s0.wp.com/latex.php?latex=%7BT%3A+X+%5Crightarrow+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%3A+X+%5Crightarrow+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%3A+X+%5Crightarrow+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T: X \rightarrow X}" class="latex" />. A point <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is said to be generic for <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> with respect to a Følner sequence <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi}" class="latex" /> if one has <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_X+f%5C+d%5Cmu+%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%7B%5Cbf+E%7D_%7Bn+%5Cin+%5CPhi_N%7D+f%28T%5En+a%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_X+f%5C+d%5Cmu+%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%7B%5Cbf+E%7D_%7Bn+%5Cin+%5CPhi_N%7D+f%28T%5En+a%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cint_X+f%5C+d%5Cmu+%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%7B%5Cbf+E%7D_%7Bn+%5Cin+%5CPhi_N%7D+f%28T%5En+a%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \int_X f\ d\mu = \lim_{N \rightarrow \infty} {\bf E}_{n \in \Phi_N} f(T^n a)" class="latex" />
for all continuous <img src="https://s0.wp.com/latex.php?latex=%7Bf%3A+X+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%3A+X+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%3A+X+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f: X \rightarrow {\bf C}}" class="latex" />. Define an (length three) dynamical Erdös progression to be a tuple <img src="https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(a,x_1,x_2)}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> with the property that there exists a sequence <img src="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_j+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_j \rightarrow \infty}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7BT%5E%7Bn_j%7D+a+%5Crightarrow+x_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%5E%7Bn_j%7D+a+%5Crightarrow+x_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%5E%7Bn_j%7D+a+%5Crightarrow+x_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T^{n_j} a \rightarrow x_1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BT%5E%7Bn_j%7D+x_1+%5Crightarrow+x_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%5E%7Bn_j%7D+x_1+%5Crightarrow+x_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%5E%7Bn_j%7D+x_1+%5Crightarrow+x_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T^{n_j} x_1 \rightarrow x_2}" class="latex" />.

Theorem <a href="#main-erdos">4</a> then follows from

<blockquote>Theorem 5 (Dynamical Erdös progression) <a name="main-dynam"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> be a countably infinite abelian group with <img src="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5BG%3A2G%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{[G:2G]}" class="latex" /> finite. Let <img src="https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X,T,\mu)}" class="latex" /> be a topological measure-preserving <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />-system, let <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> be a <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi}" class="latex" />-generic point of <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> for some Følner sequence <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi}" class="latex" />, and let <img src="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E}" class="latex" /> be a positive measure open subset of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />. Then there exists a dynamical Erdös progression <img src="https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(a,x_1,x_2)}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1 \in E}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+%5Cbigcup_%7Bt+%5Cin+G%7D+T%5Et+E%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+%5Cbigcup_%7Bt+%5Cin+G%7D+T%5Et+E%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+%5Cbigcup_%7Bt+%5Cin+G%7D+T%5Et+E%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_2 \in \bigcup_{t \in G} T^t E}" class="latex" />. </blockquote>


Indeed, we can take <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> to be <img src="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%5EG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2^G}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> to be <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T}" class="latex" /> to be the shift <img src="https://s0.wp.com/latex.php?latex=%7BT%5En+B+%3A%3D+B-n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%5En+B+%3A%3D+B-n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%5En+B+%3A%3D+B-n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T^n B := B-n}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7BE+%3A%3D+%5C%7B+B+%5Cin+2%5EG%3A+0+%5Cin+B+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE+%3A%3D+%5C%7B+B+%5Cin+2%5EG%3A+0+%5Cin+B+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE+%3A%3D+%5C%7B+B+%5Cin+2%5EG%3A+0+%5Cin+B+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E := \{ B \in 2^G: 0 \in B \}}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> to be a weak limit of the <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5CPhi_N%7D+%5Cdelta_%7BA-n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5CPhi_N%7D+%5Cdelta_%7BA-n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5CPhi_N%7D+%5Cdelta_%7BA-n%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mathop{\bf E}_{n \in \Phi_N} \delta_{A-n}}" class="latex" /> for a Følner sequence <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_N%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_N%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_N%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_N}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7B%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%7CA+%5Ccap+%5CPhi_N%7C+%2F+%7C%5CPhi_N%7C+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%7CA+%5Ccap+%5CPhi_N%7C+%2F+%7C%5CPhi_N%7C+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%7CA+%5Ccap+%5CPhi_N%7C+%2F+%7C%5CPhi_N%7C+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\lim_{N \rightarrow \infty} |A \cap \Phi_N| / |\Phi_N| > 0}" class="latex" />, at which point Theorem <a href="#main-erdos">4</a> follows from Theorem <a href="#main-dynam">5</a> after chasing definitions. (It is also possible to establish the reverse implication, but we will not need to do so here.)

A remarkable fact about this theorem is that the point <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> need not be in the support of <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" />! (In a related vein, the elements <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_j}" class="latex" /> of the Følner sequence are not required to contain the origin.)

Using a certain amount of ergodic theory and spectral theory, Kra et al. were able to reduce this theorem to a special case:

<blockquote>Theorem 6 (Reduction) <a name="main-dynam-special"></a> To prove Theorem <a href="#main-dynam">5</a>, it suffices to do so under the additional hypotheses that <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is ergodic, and there is a continuous factor map to the Kronecker factor. (In particular, the eigenfunctions of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> can be taken to be continuous.) </blockquote>


We refer the reader to the paper of Kra et al. for the details of this reduction. Now we specialize for simplicity to the case where <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5E%5Comega+%3D+%5Cbigcup_N+%7B%5Cbf+F%7D_p%5EN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5E%5Comega+%3D+%5Cbigcup_N+%7B%5Cbf+F%7D_p%5EN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5E%5Comega+%3D+%5Cbigcup_N+%7B%5Cbf+F%7D_p%5EN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G = {\bf F}_p^\omega = \bigcup_N {\bf F}_p^N}" class="latex" /> is a countable vector space over a finite field of size equal to an odd prime <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" />, so in particular <img src="https://s0.wp.com/latex.php?latex=%7B2G%3DG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2G%3DG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2G%3DG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2G=G}" class="latex" />; we also specialize to Følner sequences of the form <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+x_j+%2B+%7B%5Cbf+F%7D_p%5E%7BN_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+x_j+%2B+%7B%5Cbf+F%7D_p%5E%7BN_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+x_j+%2B+%7B%5Cbf+F%7D_p%5E%7BN_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_j = x_j + {\bf F}_p^{N_j}}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7Bx_j+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_j+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_j+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_j \in G}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BN_j+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN_j+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN_j+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N_j \geq 1}" class="latex" />. In this case we can prove a stronger statement:

<blockquote>Theorem 7 (Odd characteristic case) <a name="main-dynam-odd"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5E%5Comega%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5E%5Comega%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5E%5Comega%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G = {\bf F}_p^\omega}" class="latex" /> for an odd prime <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" />. Let <img src="https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X%2CT%2C%5Cmu%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X,T,\mu)}" class="latex" /> be a topological measure-preserving <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />-system with a continuous factor map to the Kronecker factor, and let <img src="https://s0.wp.com/latex.php?latex=%7BE_1%2C+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE_1%2C+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE_1%2C+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E_1, E_2}" class="latex" /> be open subsets of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29+%2B+%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29+%2B+%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29+%2B+%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu(E_1) + \mu(E_2) > 1}" class="latex" />. Then if <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> is a <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi}" class="latex" />-generic point of <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> for some Følner sequence <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+y_j+%2B+%7B%5Cbf+F%7D_p%5E%7Bn_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+y_j+%2B+%7B%5Cbf+F%7D_p%5E%7Bn_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+y_j+%2B+%7B%5Cbf+F%7D_p%5E%7Bn_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_j = y_j + {\bf F}_p^{n_j}}" class="latex" />, there exists an Erdös progression <img src="https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28a%2Cx_1%2Cx_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(a,x_1,x_2)}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1 \in E_1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_2 \in E_2}" class="latex" />. </blockquote>


Indeed, in the setting of Theorem <a href="#main-dynam">5</a> with the ergodicity hypothesis, the set <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbigcup_%7Bt+%5Cin+G%7D+T%5Et+E%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbigcup_%7Bt+%5Cin+G%7D+T%5Et+E%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbigcup_%7Bt+%5Cin+G%7D+T%5Et+E%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\bigcup_{t \in G} T^t E}" class="latex" /> has full measure, so the hypothesis <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2B%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2B%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2B%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu(E_1)+\mu(E_2) > 1}" class="latex" /> of Theorem <a href="#main-dynam-odd">7</a> will be verified in this case. (In the case of more general <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, this hypothesis ends up being replaced with <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2F%5BG%3A2G%5D+%2B+%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2F%5BG%3A2G%5D+%2B+%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2F%5BG%3A2G%5D+%2B+%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu(E_1)/[G:2G] + \mu(E_2) > 1}" class="latex" />; see Theorem 2.1 of <a href="https://arxiv.org/abs/2404.12201">this recent preprint of Kousek and Radic</a> for a treatment of the case <img src="https://s0.wp.com/latex.php?latex=%7BG%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%3D%7B%5Cbf+Z%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G={\bf Z}}" class="latex" /> (but the proof extends without much difficulty to the general case).)

As with Theorem <a href="#main">1</a>, Theorem <a href="#main-dynam-odd">7</a> is still an infinitary statement and does not have a direct finitary analogue (though it can likely be expressed as the conjunction of infinitely many such finitary statements, as we did with Theorem <a href="#main">1</a>). Nevertheless we can formulate the following finitary statement which can be viewed as a “baby” version of the above theorem:

<blockquote>Theorem 8 (Finitary model problem) <a name="model"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BX+%3D+%28X%2Cd%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX+%3D+%28X%2Cd%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX+%3D+%28X%2Cd%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X = (X,d)}" class="latex" /> be a compact metric space, let <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5EN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5EN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D+%7B%5Cbf+F%7D_p%5EN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G = {\bf F}_p^N}" class="latex" /> be a finite vector space over a field of odd prime order. Let <img src="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T}" class="latex" /> be an action of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> on <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> by homeomorphisms, let <img src="https://s0.wp.com/latex.php?latex=%7Ba+%5Cin+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba+%5Cin+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba+%5Cin+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a \in X}" class="latex" />, and let <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> be the associated <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />-invariant measure <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu+%3D+%7B%5Cbf+E%7D_%7Bx+%5Cin+G%7D+%5Cdelta_%7BT%5Ex+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu+%3D+%7B%5Cbf+E%7D_%7Bx+%5Cin+G%7D+%5Cdelta_%7BT%5Ex+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu+%3D+%7B%5Cbf+E%7D_%7Bx+%5Cin+G%7D+%5Cdelta_%7BT%5Ex+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu = {\bf E}_{x \in G} \delta_{T^x a}}" class="latex" />. Let <img src="https://s0.wp.com/latex.php?latex=%7BE_1%2C+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BE_1%2C+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BE_1%2C+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{E_1, E_2}" class="latex" /> be subsets of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29+%2B+%5Cmu%28E_2%29+%3E+1+%2B+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29+%2B+%5Cmu%28E_2%29+%3E+1+%2B+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29+%2B+%5Cmu%28E_2%29+%3E+1+%2B+%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu(E_1) + \mu(E_2) > 1 + \delta}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cdelta%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cdelta%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\delta>0}" class="latex" />. Then for any <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon>0}" class="latex" />, there exist <img src="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%2C+x_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%2C+x_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%2C+x_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1 \in E_1, x_2 \in E_2}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+G%3A+d%28T%5Eh+a%2Cx_1%29+%5Cleq+%5Cvarepsilon%2C+d%28T%5Eh+x_1%2Cx_2%29+%5Cleq+%5Cvarepsilon+%5C%7D%7C+%5Cgg_%7Bp%2C%5Cdelta%2C%5Cvarepsilon%2CX%7D+%7CG%7C.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+G%3A+d%28T%5Eh+a%2Cx_1%29+%5Cleq+%5Cvarepsilon%2C+d%28T%5Eh+x_1%2Cx_2%29+%5Cleq+%5Cvarepsilon+%5C%7D%7C+%5Cgg_%7Bp%2C%5Cdelta%2C%5Cvarepsilon%2CX%7D+%7CG%7C.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+G%3A+d%28T%5Eh+a%2Cx_1%29+%5Cleq+%5Cvarepsilon%2C+d%28T%5Eh+x_1%2Cx_2%29+%5Cleq+%5Cvarepsilon+%5C%7D%7C+%5Cgg_%7Bp%2C%5Cdelta%2C%5Cvarepsilon%2CX%7D+%7CG%7C.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ h \in G: d(T^h a,x_1) \leq \varepsilon, d(T^h x_1,x_2) \leq \varepsilon \}| \gg_{p,\delta,\varepsilon,X} |G|." class="latex" />
</blockquote>


The important thing here is that the bounds are uniform in the dimension <img src="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N}" class="latex" /> (as well as the initial point <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> and the action <img src="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T}" class="latex" />).

Let us now give a finitary proof of Theorem <a href="#model">8</a>. We can cover the compact metric space <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> by a finite collection <img src="https://s0.wp.com/latex.php?latex=%7BB_1%2C%5Cdots%2CB_M%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB_1%2C%5Cdots%2CB_M%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB_1%2C%5Cdots%2CB_M%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B_1,\dots,B_M}" class="latex" /> of open balls of radius <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon/2}" class="latex" />. This induces a coloring function <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%3A+X+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%3A+X+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%3A+X+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\tilde c: X \rightarrow \{1,\dots,M\}}" class="latex" /> that assigns to each point in <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> the index <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" /> of the first ball <img src="https://s0.wp.com/latex.php?latex=%7BB_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BB_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BB_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{B_m}" class="latex" /> that covers that point. This then induces a coloring <img src="https://s0.wp.com/latex.php?latex=%7Bc%3A+G+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%3A+G+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%3A+G+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c: G \rightarrow \{1,\dots,M\}}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> by the formula <img src="https://s0.wp.com/latex.php?latex=%7Bc%28h%29+%3A%3D+%5Ctilde+c%28T%5Eh+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%28h%29+%3A%3D+%5Ctilde+c%28T%5Eh+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%28h%29+%3A%3D+%5Ctilde+c%28T%5Eh+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c(h) := \tilde c(T^h a)}" class="latex" />. We also define the pullbacks <img src="https://s0.wp.com/latex.php?latex=%7BA_i+%3A%3D+%5C%7B+h+%5Cin+G%3A+T%5Eh+a+%5Cin+E_i+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA_i+%3A%3D+%5C%7B+h+%5Cin+G%3A+T%5Eh+a+%5Cin+E_i+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA_i+%3A%3D+%5C%7B+h+%5Cin+G%3A+T%5Eh+a+%5Cin+E_i+%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A_i := \{ h \in G: T^h a \in E_i \}}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i=1,2}" class="latex" />. By hypothesis, we have <img src="https://s0.wp.com/latex.php?latex=%7B%7CA_1%7C+%2B+%7CA_2%7C+%3E+%281%2B%5Cdelta%29%7CG%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA_1%7C+%2B+%7CA_2%7C+%3E+%281%2B%5Cdelta%29%7CG%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA_1%7C+%2B+%7CA_2%7C+%3E+%281%2B%5Cdelta%29%7CG%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A_1| + |A_2| > (1+\delta)|G|}" class="latex" />, and it will now suffice by the triangle inequality to show that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+G%3A+c%28h%29+%3D+c%28x_1%29%3B+c%28h%2Bx_1%29%3Dc%28x_2%29+%5C%7D%7C+%5Cgg_%7Bp%2C%5Cdelta%2CM%7D+%7CG%7C.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+G%3A+c%28h%29+%3D+c%28x_1%29%3B+c%28h%2Bx_1%29%3Dc%28x_2%29+%5C%7D%7C+%5Cgg_%7Bp%2C%5Cdelta%2CM%7D+%7CG%7C.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+G%3A+c%28h%29+%3D+c%28x_1%29%3B+c%28h%2Bx_1%29%3Dc%28x_2%29+%5C%7D%7C+%5Cgg_%7Bp%2C%5Cdelta%2CM%7D+%7CG%7C.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ h \in G: c(h) = c(x_1); c(h+x_1)=c(x_2) \}| \gg_{p,\delta,M} |G|." class="latex" />
Now we apply the <a href="https://arxiv.org/abs/1002.2028">arithmetic lemma of Green</a> with some regularity parameter <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa>0}" class="latex" /> to be chosen later. This allows us to partition <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> into cosets of a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of index <img src="https://s0.wp.com/latex.php?latex=%7BO_%7Bp%2C%5Ckappa%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO_%7Bp%2C%5Ckappa%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO_%7Bp%2C%5Ckappa%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O_{p,\kappa}(1)}" class="latex" />, such that on all but <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa+%5BG%3AH%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa+%5BG%3AH%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa+%5BG%3AH%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa [G:H]}" class="latex" /> of these cosets <img src="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y+H}" class="latex" />, all the color classes <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7Bx+%5Cin+y%2BH%3A+c%28x%29+%3D+c_0%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7Bx+%5Cin+y%2BH%3A+c%28x%29+%3D+c_0%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7Bx+%5Cin+y%2BH%3A+c%28x%29+%3D+c_0%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\{x \in y+H: c(x) = c_0\}}" class="latex" /> are <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%5E%7B100%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%5E%7B100%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%5E%7B100%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa^{100}}" class="latex" />-regular in the Fourier (<img src="https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U^2}" class="latex" />) sense. Now we sample <img src="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1}" class="latex" /> uniformly from <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, and set <img src="https://s0.wp.com/latex.php?latex=%7Bx_2+%3A%3D+2x_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_2+%3A%3D+2x_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_2+%3A%3D+2x_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_2 := 2x_1}" class="latex" />; as <img src="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p}" class="latex" /> is odd, <img src="https://s0.wp.com/latex.php?latex=%7Bx_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_2}" class="latex" /> is also uniform in <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />. If <img src="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1}" class="latex" /> lies in a coset <img src="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y+H}" class="latex" />, then <img src="https://s0.wp.com/latex.php?latex=%7Bx_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_2}" class="latex" /> will lie in <img src="https://s0.wp.com/latex.php?latex=%7B2y%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2y%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2y%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2y+H}" class="latex" />. By removing an exceptional event of probability <img src="https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(\kappa)}" class="latex" />, we may assume that neither of these cosetgs <img src="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y+H}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7B2y%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2y%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2y%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2y+H}" class="latex" /> is a bad coset. By removing a further exceptional event of probability <img src="https://s0.wp.com/latex.php?latex=%7BO_M%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO_M%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO_M%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O_M(\kappa)}" class="latex" />, we may also assume that <img src="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1}" class="latex" /> is in a popular color class of <img src="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y+H}" class="latex" /> in the sense that <a name="X1"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+x+%5Cin+y%2BH%3A+c%28x%29+%3D+c%28x_1%29+%5C%7D%7C+%5Cgeq+%5Ckappa+%7CH%7C+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+x+%5Cin+y%2BH%3A+c%28x%29+%3D+c%28x_1%29+%5C%7D%7C+%5Cgeq+%5Ckappa+%7CH%7C+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+x+%5Cin+y%2BH%3A+c%28x%29+%3D+c%28x_1%29+%5C%7D%7C+%5Cgeq+%5Ckappa+%7CH%7C+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ x \in y+H: c(x) = c(x_1) \}| \geq \kappa |H| \ \ \ \ \ (1)" class="latex" />
</a> since the set of exceptional <img src="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1}" class="latex" /> that fail to achieve this only are hit with probability <img src="https://s0.wp.com/latex.php?latex=%7BO%28M%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28M%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28M%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(M\kappa)}" class="latex" />. Similarly we may assume that <a name="X2"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+x+%5Cin+2y%2BH%3A+c%28x%29+%3D+c%28x_2%29+%5C%7D%7C+%5Cgeq+%5Ckappa+%7CH%7C.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+x+%5Cin+2y%2BH%3A+c%28x%29+%3D+c%28x_2%29+%5C%7D%7C+%5Cgeq+%5Ckappa+%7CH%7C.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+x+%5Cin+2y%2BH%3A+c%28x%29+%3D+c%28x_2%29+%5C%7D%7C+%5Cgeq+%5Ckappa+%7CH%7C.+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ x \in 2y+H: c(x) = c(x_2) \}| \geq \kappa |H|. \ \ \ \ \ (2)" class="latex" />
</a> Now we consider the quantity <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+y%2BH%3A+c%28h%29+%3D+c%28x_1%29%3B+c%28h%2Bx_1%29%3Dc%28x_2%29+%5C%7D%7C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+y%2BH%3A+c%28h%29+%3D+c%28x_1%29%3B+c%28h%2Bx_1%29%3Dc%28x_2%29+%5C%7D%7C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+y%2BH%3A+c%28h%29+%3D+c%28x_1%29%3B+c%28h%2Bx_1%29%3Dc%28x_2%29+%5C%7D%7C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ h \in y+H: c(h) = c(x_1); c(h+x_1)=c(x_2) \}|" class="latex" />
which we can write as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7CH%7C+%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_1%29%29%7D%28h%29+1_%7Bc%5E%7B-1%7D%28c%28x_2%29%29%7D%28h%2Bx_1%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7CH%7C+%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_1%29%29%7D%28h%29+1_%7Bc%5E%7B-1%7D%28c%28x_2%29%29%7D%28h%2Bx_1%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7CH%7C+%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_1%29%29%7D%28h%29+1_%7Bc%5E%7B-1%7D%28c%28x_2%29%29%7D%28h%2Bx_1%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |H| {\bf E}_{h \in y+H} 1_{c^{-1}(c(x_1))}(h) 1_{c^{-1}(c(x_2))}(h+x_1)." class="latex" />
Both factors here are <img src="https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%5E%7B100%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%5E%7B100%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%5E%7B100%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(\kappa^{100})}" class="latex" />-uniform in their respective cosets. Thus by standard Fourier calculations, we see that after excluding another exceptional event of probabitiy <img src="https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28%5Ckappa%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(\kappa)}" class="latex" />, this quantity is equal to <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7CH%7C+%28%28%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_1%29%29%7D%28h%29%29+%28%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_2%29%29%7D%28h%2Bx_1%29%29+%2B+O%28%5Ckappa%5E%7B10%7D%29%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7CH%7C+%28%28%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_1%29%29%7D%28h%29%29+%28%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_2%29%29%7D%28h%2Bx_1%29%29+%2B+O%28%5Ckappa%5E%7B10%7D%29%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7CH%7C+%28%28%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_1%29%29%7D%28h%29%29+%28%7B%5Cbf+E%7D_%7Bh+%5Cin+y%2BH%7D+1_%7Bc%5E%7B-1%7D%28c%28x_2%29%29%7D%28h%2Bx_1%29%29+%2B+O%28%5Ckappa%5E%7B10%7D%29%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |H| (({\bf E}_{h \in y+H} 1_{c^{-1}(c(x_1))}(h)) ({\bf E}_{h \in y+H} 1_{c^{-1}(c(x_2))}(h+x_1)) + O(\kappa^{10}))." class="latex" />
By <a href="#X1">(1)</a>, <a href="#X2">(2)</a>, this expression is <img src="https://s0.wp.com/latex.php?latex=%7B%5Cgg+%5Ckappa%5E2+%7CH%7C+%5Cgg_%7Bp%2C%5Ckappa%7D+%7CG%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cgg+%5Ckappa%5E2+%7CH%7C+%5Cgg_%7Bp%2C%5Ckappa%7D+%7CG%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cgg+%5Ckappa%5E2+%7CH%7C+%5Cgg_%7Bp%2C%5Ckappa%7D+%7CG%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\gg \kappa^2 |H| \gg_{p,\kappa} |G|}" class="latex" />. By choosing <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa}" class="latex" /> small enough depending on <img src="https://s0.wp.com/latex.php?latex=%7BM%2C%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BM%2C%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BM%2C%5Cdelta%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{M,\delta}" class="latex" />, we can ensure that <img src="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1+%5Cin+E_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1 \in E_1}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_2+%5Cin+E_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_2 \in E_2}" class="latex" />, and the claim follows.

Now we can prove the infinitary result in Theorem <a href="#main-dynam-odd">7</a>. Let us place a metric <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" /> on <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />. By sparsifying the Følner sequence <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+y_j+%2B+%7B%5Cbf+F%7D_p%5E%7BN_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+y_j+%2B+%7B%5Cbf+F%7D_p%5E%7BN_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_j+%3D+y_j+%2B+%7B%5Cbf+F%7D_p%5E%7BN_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_j = y_j + {\bf F}_p^{N_j}}" class="latex" />, we may assume that the <img src="https://s0.wp.com/latex.php?latex=%7Bn_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_j}" class="latex" /> grow as fast as we wish. Once we do so, we claim that for each <img src="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{J}" class="latex" />, we can find <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D+%5Cin+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D+%5Cin+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D+%5Cin+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{1,J}, x_{2,J} \in X}" class="latex" /> such that for each <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq j \leq J}" class="latex" />, there exists <img src="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Cin+%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_j+%5Cin+%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_j+%5Cin+%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_j \in \Phi_j}" class="latex" /> that lies outside of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_p^j}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28T%5E%7Bn_j%7D+a%2C+x_%7B1%2CJ%7D%29+%5Cleq+1%2Fj%2C+%5Cquad+d%28T%5E%7Bn_j%7D+x_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D%29+%5Cleq+1%2Fj.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28T%5E%7Bn_j%7D+a%2C+x_%7B1%2CJ%7D%29+%5Cleq+1%2Fj%2C+%5Cquad+d%28T%5E%7Bn_j%7D+x_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D%29+%5Cleq+1%2Fj.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28T%5E%7Bn_j%7D+a%2C+x_%7B1%2CJ%7D%29+%5Cleq+1%2Fj%2C+%5Cquad+d%28T%5E%7Bn_j%7D+x_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D%29+%5Cleq+1%2Fj.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle d(T^{n_j} a, x_{1,J}) \leq 1/j, \quad d(T^{n_j} x_{1,J}, x_{2,J}) \leq 1/j." class="latex" />
Passing to a subsequence to make <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%2C+x_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{1,J}, x_{2,J}}" class="latex" /> converge to <img src="https://s0.wp.com/latex.php?latex=%7Bx_1%2C+x_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_1%2C+x_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_1%2C+x_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_1, x_2}" class="latex" /> respectively, we obtain the desired Erdös progression.

Fix <img src="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{J}" class="latex" />, and let <img src="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{M}" class="latex" /> be a large parameter (much larger than <img src="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BJ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{J}" class="latex" />) to be chosen later. By genericity, we know that the discrete measures <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_M%7D+%5Cdelta_%7BT%5Eh+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_M%7D+%5Cdelta_%7BT%5Eh+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_M%7D+%5Cdelta_%7BT%5Eh+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf E}_{h \in \Phi_M} \delta_{T^h a}}" class="latex" /> converge vaguely to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" />, so any point in the support in <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> can be approximated by some point <img src="https://s0.wp.com/latex.php?latex=%7BT%5Eh+a%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%5Eh+a%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%5Eh+a%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T^h a}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h \in \Phi_M}" class="latex" />. Unfortunately, <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" /> does not necessarily lie in this support! (Note that <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_M}" class="latex" /> need not contain the origin.) However, we are assuming a continuous factor map <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3AX+%5Crightarrow+Z%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3AX+%5Crightarrow+Z%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%3AX+%5Crightarrow+Z%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi:X \rightarrow Z}" class="latex" /> to the Kronecker factor <img src="https://s0.wp.com/latex.php?latex=%7BZ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BZ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BZ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Z}" class="latex" />, which is a compact abelian group, and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" /> pushes down to the Haar measure of <img src="https://s0.wp.com/latex.php?latex=%7BZ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BZ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BZ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Z}" class="latex" />, which has full support. In particular, thus pushforward contains <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi(a)}" class="latex" />. As a consequence, we can find <img src="https://s0.wp.com/latex.php?latex=%7Bh_M+%5Cin+%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh_M+%5Cin+%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh_M+%5Cin+%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h_M \in \Phi_M}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28T%5E%7Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28T%5E%7Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%28T%5E%7Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi(T^{h_M} a)}" class="latex" /> converges to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%28a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi(a)}" class="latex" />, even if we cannot ensure that <img src="https://s0.wp.com/latex.php?latex=%7BT%5E%7Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%5E%7Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%5E%7Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T^{h_M} a}" class="latex" /> converges to <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" />. We are assuming that <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_M%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_M}" class="latex" /> is a coset of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_p^{n_M}}" class="latex" />, so now <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+%5Cdelta_%7BT%5E%7Bh%2Bh_M%7D+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+%5Cdelta_%7BT%5E%7Bh%2Bh_M%7D+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+%5Cdelta_%7BT%5E%7Bh%2Bh_M%7D+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf E}_{h \in {\bf F}_p^{n_M}} \delta_{T^{h+h_M} a}}" class="latex" /> converges vaguely to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu}" class="latex" />.

We make the random choice <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D+%3A%3D+T%5E%7Bh_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D+%3A%3D+T%5E%7Bh_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D+%3A%3D+T%5E%7Bh_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{1,J} := T^{h_*+h_M} a}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D+%3A%3D+T%5E%7B2h_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D+%3A%3D+T%5E%7B2h_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D+%3A%3D+T%5E%7B2h_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{2,J} := T^{2h_*+h_M} a}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h_*}" class="latex" /> is drawn uniformly at random from <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_p^{n_M}}" class="latex" />. This is not the only possible choice that can be made here, and is in fact not optimal in certain respects (in particular, it creates a fair bit of coupling between <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{1,J}}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{2,J}}" class="latex" />), but is easy to describe and will suffice for our argument. (A more appropriate choice, closer to the arguments of Kra et al., would be to <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{2,J}}" class="latex" /> in the above construction by <img src="https://s0.wp.com/latex.php?latex=%7BT%5E%7B2h_%2A%2Bk_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%5E%7B2h_%2A%2Bk_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%5E%7B2h_%2A%2Bk_%2A%2Bh_M%7D+a%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T^{2h_*+k_*+h_M} a}" class="latex" />, where the additional shift <img src="https://s0.wp.com/latex.php?latex=%7Bk_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k_*}" class="latex" /> is a random variable in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_p^{n_M}}" class="latex" /> independent of <img src="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h_*}" class="latex" /> that is uniformly drawn from all shifts annihilated by the first <img src="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{M}" class="latex" /> characters associated to some enumeration of the (necessarily countable) point spectrum of <img src="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T}" class="latex" />, but this is harder to describe.)

Since we are in odd characteristic, the map <img src="https://s0.wp.com/latex.php?latex=%7Bh+%5Cmapsto+2h%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh+%5Cmapsto+2h%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh+%5Cmapsto+2h%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h \mapsto 2h}" class="latex" /> is a permutation on <img src="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h \in {\bf F}_p^{n_M}}" class="latex" />, and so <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{1,J}}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{2,J}}" class="latex" /> are both distributed according to the law <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+%5Cdelta_%7BT%5E%7Bh%2Bh_M%7D+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+%5Cdelta_%7BT%5E%7Bh%2Bh_M%7D+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+E%7D_%7Bh+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+%5Cdelta_%7BT%5E%7Bh%2Bh_M%7D+a%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf E}_{h \in {\bf F}_p^{n_M}} \delta_{T^{h+h_M} a}}" class="latex" />, though they are coupled to each other. In particular, by vague convergence (and inner regularity) we have <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B1%2CJ%7D+%5Cin+E_1+%29+%5Cgeq+%5Cmu%28E_1%29+-+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B1%2CJ%7D+%5Cin+E_1+%29+%5Cgeq+%5Cmu%28E_1%29+-+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B1%2CJ%7D+%5Cin+E_1+%29+%5Cgeq+%5Cmu%28E_1%29+-+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf P}( x_{1,J} \in E_1 ) \geq \mu(E_1) - o(1)" class="latex" />
and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B2%2CJ%7D+%5Cin+E_2+%29+%5Cgeq+%5Cmu%28E_2%29+-+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B2%2CJ%7D+%5Cin+E_2+%29+%5Cgeq+%5Cmu%28E_2%29+-+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B2%2CJ%7D+%5Cin+E_2+%29+%5Cgeq+%5Cmu%28E_2%29+-+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf P}( x_{2,J} \in E_2 ) \geq \mu(E_2) - o(1)" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{o(1)}" class="latex" /> denotes a quantity that goes to zero as <img src="https://s0.wp.com/latex.php?latex=%7BM+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BM+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BM+%5Crightarrow+%5Cinfty%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{M \rightarrow \infty}" class="latex" /> (holding all other parameters fixed). By the hypothesis <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2B%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2B%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmu%28E_1%29%2B%5Cmu%28E_2%29+%3E+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\mu(E_1)+\mu(E_2) > 1}" class="latex" />, we thus have <a name="x1x2"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B1%2CJ%7D+%5Cin+E_1%2C+x_%7B2%2CJ%7D+%5Cin+E_2+%29+%5Cgeq+%5Ckappa+-+o%281%29+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B1%2CJ%7D+%5Cin+E_1%2C+x_%7B2%2CJ%7D+%5Cin+E_2+%29+%5Cgeq+%5Ckappa+-+o%281%29+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+P%7D%28+x_%7B1%2CJ%7D+%5Cin+E_1%2C+x_%7B2%2CJ%7D+%5Cin+E_2+%29+%5Cgeq+%5Ckappa+-+o%281%29+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf P}( x_{1,J} \in E_1, x_{2,J} \in E_2 ) \geq \kappa - o(1) \ \ \ \ \ (3)" class="latex" />
</a> for some <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa>0}" class="latex" /> independent of <img src="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{M}" class="latex" />.

We will show that for each <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq j \leq J}" class="latex" />, one has <a name="hphi"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j%3A+d%28T%5E%7Bh%7D+a%2Cx_%7B1%2CJ%7D%29+%5Cleq+1%2Fj%2C+d%28T%5Eh+x_%7B1%2CJ%7D%2Cx_%7B2%2CJ%7D%29+%5Cleq+1%2Fj+%5C%7D%7C+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j%3A+d%28T%5E%7Bh%7D+a%2Cx_%7B1%2CJ%7D%29+%5Cleq+1%2Fj%2C+d%28T%5Eh+x_%7B1%2CJ%7D%2Cx_%7B2%2CJ%7D%29+%5Cleq+1%2Fj+%5C%7D%7C+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j%3A+d%28T%5E%7Bh%7D+a%2Cx_%7B1%2CJ%7D%29+%5Cleq+1%2Fj%2C+d%28T%5Eh+x_%7B1%2CJ%7D%2Cx_%7B2%2CJ%7D%29+%5Cleq+1%2Fj+%5C%7D%7C+%5C+%5C+%5C+%5C+%5C+%284%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ h \in \Phi_j: d(T^{h} a,x_{1,J}) \leq 1/j, d(T^h x_{1,J},x_{2,J}) \leq 1/j \}| \ \ \ \ \ (4)" class="latex" />
</a> <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cgg_%7Bp%2C%5Ckappa%2Cj%2CX%7D+%281-o%281%29%29+%7C%5CPhi_j%7C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cgg_%7Bp%2C%5Ckappa%2Cj%2CX%7D+%281-o%281%29%29+%7C%5CPhi_j%7C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cgg_%7Bp%2C%5Ckappa%2Cj%2CX%7D+%281-o%281%29%29+%7C%5CPhi_j%7C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \gg_{p,\kappa,j,X} (1-o(1)) |\Phi_j|" class="latex" />
outside of an event of probability at most <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F2%5E%7Bj%2B1%7D%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F2%5E%7Bj%2B1%7D%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F2%5E%7Bj%2B1%7D%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa/2^{j+1}+o(1)}" class="latex" /> (compare with Theorem <a href="#model">8</a>). If this is the case, then by the union bound we can find (for <img src="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{M}" class="latex" /> large enough) a choice of <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B1%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{1,J}}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx_%7B2%2CJ%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x_{2,J}}" class="latex" /> obeying <a href="#x1x2">(3)</a> as well as <a href="#hphi">(4)</a> for all <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq j \leq J}" class="latex" />. If the <img src="https://s0.wp.com/latex.php?latex=%7BN_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BN_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BN_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{N_j}" class="latex" /> grow fast enough, we can then ensure that for each <img src="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1+%5Cleq+j+%5Cleq+J%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1 \leq j \leq J}" class="latex" /> one can find (again for <img src="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BM%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{M}" class="latex" /> large enough) <img src="https://s0.wp.com/latex.php?latex=%7Bn_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n_j}" class="latex" /> in the set in <a href="#hphi">(4)</a> that avoids <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5Ej%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_p^j}" class="latex" />, and the claim follows.

It remains to show <a href="#hphi">(4)</a> outside of an exceptional event of acceptable probability. Let <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%3A+X+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM_j%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%3A+X+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM_j%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%3A+X+%5Crightarrow+%5C%7B1%2C%5Cdots%2CM_j%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\tilde c: X \rightarrow \{1,\dots,M_j\}}" class="latex" /> be the coloring function from the proof of Theorem <a href="#model">8</a> (with <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+%3A%3D+1%2Fj%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+%3A%3D+1%2Fj%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+%3A%3D+1%2Fj%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon := 1/j}" class="latex" />). Then it suffices to show that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j%3A+c_0%28h%29+%3D+c%28h_%2A%29%3B+c%28h%2Bh_%2A%29%3Dc%282h_%2A%29+%5C%7D%7C+%5Cgg_%7Bp%2C%5Ckappa%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j%7C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j%3A+c_0%28h%29+%3D+c%28h_%2A%29%3B+c%28h%2Bh_%2A%29%3Dc%282h_%2A%29+%5C%7D%7C+%5Cgg_%7Bp%2C%5Ckappa%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j%7C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j%3A+c_0%28h%29+%3D+c%28h_%2A%29%3B+c%28h%2Bh_%2A%29%3Dc%282h_%2A%29+%5C%7D%7C+%5Cgg_%7Bp%2C%5Ckappa%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j%7C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ h \in \Phi_j: c_0(h) = c(h_*); c(h+h_*)=c(2h_*) \}| \gg_{p,\kappa,M_j} (1-o(1)) |\Phi_j|" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7Bc_0%28h%29+%3A%3D+%5Ctilde+c%28T%5Eh+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc_0%28h%29+%3A%3D+%5Ctilde+c%28T%5Eh+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc_0%28h%29+%3A%3D+%5Ctilde+c%28T%5Eh+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c_0(h) := \tilde c(T^h a)}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bc%28h%29+%3A%3D+%5Ctilde+c%28T%5E%7Bh%2Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%28h%29+%3A%3D+%5Ctilde+c%28T%5E%7Bh%2Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%28h%29+%3A%3D+%5Ctilde+c%28T%5E%7Bh%2Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c(h) := \tilde c(T^{h+h_M} a)}" class="latex" />. This is a counting problem associated to the patterm <img src="https://s0.wp.com/latex.php?latex=%7B%28h_%2A%2C+h%2C+h%2Bh_%2A%2C+2h_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28h_%2A%2C+h%2C+h%2Bh_%2A%2C+2h_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28h_%2A%2C+h%2C+h%2Bh_%2A%2C+2h_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(h_*, h, h+h_*, 2h_*)}" class="latex" />; if we concatenate the <img src="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h_*}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B2h_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2h_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2h_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2h_*}" class="latex" /> components of the pattern, this is a classic “complexity one” pattern, of the type that would be expected to be amenable to Fourier analysis (especially if one applies Cauchy-Schwarz to eliminate the <img src="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h_*}" class="latex" /> averaging and absolute value, at which point one is left with the <img src="https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U^2}" class="latex" /> pattern <img src="https://s0.wp.com/latex.php?latex=%7B%28h%2C+h%2Bh_%2A%2C+h%27%2C+h%27%2Bh_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28h%2C+h%2Bh_%2A%2C+h%27%2C+h%27%2Bh_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28h%2C+h%2Bh_%2A%2C+h%27%2C+h%27%2Bh_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(h, h+h_*, h', h'+h_*)}" class="latex" />).

In the finitary setting, we used the arithmetic regularity lemma. Here, we will need to use the Kronecker factor instead. The indicator function <img src="https://s0.wp.com/latex.php?latex=%7B1_%7B%5Ctilde+c%5E%7B-1%7D%28i%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1_%7B%5Ctilde+c%5E%7B-1%7D%28i%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1_%7B%5Ctilde+c%5E%7B-1%7D%28i%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1_{\tilde c^{-1}(i)}}" class="latex" /> of a level set of the coloring function <img src="https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ctilde+c%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\tilde c}" class="latex" /> is a bounded measurable function of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />, and can thus be decomposed into a function <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i}" class="latex" /> that is measurable on the Kronecker factor, plus an error term <img src="https://s0.wp.com/latex.php?latex=%7Bg_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g_i}" class="latex" /> that is orthogonal to that factor and thus is weakly mixing in the sense that <img src="https://s0.wp.com/latex.php?latex=%7B%7C%5Clangle+T%5Eh+g_i%2C+g_i+%5Crangle%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7C%5Clangle+T%5Eh+g_i%2C+g_i+%5Crangle%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7C%5Clangle+T%5Eh+g_i%2C+g_i+%5Crangle%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|\langle T^h g_i, g_i \rangle|}" class="latex" /> tends to zero on average (or equivalently, that the Host-Kra seminorm <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7Cg_i%5C%7C_%7BU%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7Cg_i%5C%7C_%7BU%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7Cg_i%5C%7C_%7BU%5E2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\|g_i\|_{U^2}}" class="latex" /> vanishes). Meanwhile, for any <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon+%3E+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon > 0}" class="latex" />, the Kronecker-measurable function <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i}" class="latex" /> can be decomposed further as <img src="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D+%2B+k_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D+%2B+k_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D+%2B+k_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P_{i,\varepsilon} + k_{i,\varepsilon}}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P_{i,\varepsilon}}" class="latex" /> is a bounded “trigonometric polynomial” (a finite sum of eigenfunctions) and <img src="https://s0.wp.com/latex.php?latex=%7B%5C%7Ck_%7Bi%2C%5Cvarepsilon%7D%5C%7C_%7BL%5E2%7D+%3C+%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5C%7Ck_%7Bi%2C%5Cvarepsilon%7D%5C%7C_%7BL%5E2%7D+%3C+%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5C%7Ck_%7Bi%2C%5Cvarepsilon%7D%5C%7C_%7BL%5E2%7D+%3C+%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\|k_{i,\varepsilon}\|_{L^2} < \varepsilon}" class="latex" />. The polynomial <img src="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP_%7Bi%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P_{i,\varepsilon}}" class="latex" /> is continuous by hypothesis. The other two terms in the decomposition are merely meaurable, but can be approximated to arbitrary accuracy by continuous functions. The upshot is that we can arrive at a decomposition <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7B%5Ctilde+c%5E%7B-1%7D%28i%29%7D+%3D+P_%7Bi%2C%5Cvarepsilon%7D+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7B%5Ctilde+c%5E%7B-1%7D%28i%29%7D+%3D+P_%7Bi%2C%5Cvarepsilon%7D+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7B%5Ctilde+c%5E%7B-1%7D%28i%29%7D+%3D+P_%7Bi%2C%5Cvarepsilon%7D+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 1_{\tilde c^{-1}(i)} = P_{i,\varepsilon} + k_{i,\varepsilon,\varepsilon'} + g_{i,\varepsilon'}" class="latex" />
(analogous to the arithmetic regularity lemma) for any <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2C%5Cvarepsilon%27%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2C%5Cvarepsilon%27%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2C%5Cvarepsilon%27%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon,\varepsilon'>0}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7Bk_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k_{i,\varepsilon,\varepsilon'}}" class="latex" /> is a bounded continuous function of <img src="https://s0.wp.com/latex.php?latex=%7BL%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BL%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BL%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{L^2}" class="latex" /> norm at most <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g_{i,\varepsilon'}}" class="latex" /> is a bounded continuous function of <img src="https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U^2}" class="latex" /> norm at most <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon'}" class="latex" /> (in practice we will take <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon'}" class="latex" /> much smaller than <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon}" class="latex" />). Pulling back to <img src="https://s0.wp.com/latex.php?latex=%7Bc%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c}" class="latex" />, we then have <a name="chi"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7Bc%28h%29%3Di%7D+%3D+P_%7Bi%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_M%7D+a%29+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_M%7Da%29+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_M%7Da%29.+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7Bc%28h%29%3Di%7D+%3D+P_%7Bi%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_M%7D+a%29+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_M%7Da%29+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_M%7Da%29.+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7Bc%28h%29%3Di%7D+%3D+P_%7Bi%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_M%7D+a%29+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_M%7Da%29+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_M%7Da%29.+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 1_{c(h)=i} = P_{i,\varepsilon}(T^{h+h_M} a) + k_{i,\varepsilon,\varepsilon'}(T^{h+h_M}a) + g_{i,\varepsilon'}(T^{h+h_M}a). \ \ \ \ \ (5)" class="latex" />
</a> Let <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2C%5Cvarepsilon%27%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2C%5Cvarepsilon%27%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%2C%5Cvarepsilon%27%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon,\varepsilon'>0}" class="latex" /> be chosen later. The trigonometric polynomial <img src="https://s0.wp.com/latex.php?latex=%7Bh+%5Cmapsto+P_%7Bi%2C%5Cvarepsilon%7D%28T%5E%7Bh%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh+%5Cmapsto+P_%7Bi%2C%5Cvarepsilon%7D%28T%5E%7Bh%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh+%5Cmapsto+P_%7Bi%2C%5Cvarepsilon%7D%28T%5E%7Bh%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h \mapsto P_{i,\varepsilon}(T^{h} a)}" class="latex" /> is just a sum of <img src="https://s0.wp.com/latex.php?latex=%7BO_%7B%5Cvarepsilon%2CM_j%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO_%7B%5Cvarepsilon%2CM_j%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO_%7B%5Cvarepsilon%2CM_j%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O_{\varepsilon,M_j}(1)}" class="latex" /> characters on <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, so one can find a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> of index <img src="https://s0.wp.com/latex.php?latex=%7BO_%7Bp%2C%5Cvarepsilon%2CM_j%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO_%7Bp%2C%5Cvarepsilon%2CM_j%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO_%7Bp%2C%5Cvarepsilon%2CM_j%7D%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O_{p,\varepsilon,M_j}(1)}" class="latex" /> such that these polynomial are constant on each coset of <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h_*}" class="latex" /> lies in some coset <img src="https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a_*+H}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B2h_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2h_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2h_%2A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2h_*}" class="latex" /> lies in the coset <img src="https://s0.wp.com/latex.php?latex=%7B2a_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2a_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2a_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2a_*+H}" class="latex" />. We then restrict <img src="https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h}" class="latex" /> to also lie in <img src="https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a_*+H}" class="latex" />, and we will show that <a name="targ"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%3A+c_0%28h%29+%3D+c%28h_%2A%29%3B+c%28h%2Bh_%2A%29%3Dc%282h_%2A%29+%5C%7D%7C+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%3A+c_0%28h%29+%3D+c%28h_%2A%29%3B+c%28h%2Bh_%2A%29%3Dc%282h_%2A%29+%5C%7D%7C+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5C%7B+h+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%3A+c_0%28h%29+%3D+c%28h_%2A%29%3B+c%28h%2Bh_%2A%29%3Dc%282h_%2A%29+%5C%7D%7C+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle |\{ h \in \Phi_j \cap (a_*+H): c_0(h) = c(h_*); c(h+h_*)=c(2h_*) \}| \ \ \ \ \ (6)" class="latex" />
</a> <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \gg_{\kappa,p,M_j} (1-o(1)) |\Phi_j \cap (a_*+H)| " class="latex" />
outside of an exceptional event of proability <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F2%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F2%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F2%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa/2+o(1)}" class="latex" />, which will establish our claim because <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon}" class="latex" /> will ultimately be chosen to dependon <img src="https://s0.wp.com/latex.php?latex=%7Bp%2C%5Ckappa%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%2C%5Ckappa%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%2C%5Ckappa%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p,\kappa,M_j}" class="latex" />.

The left-hand side can be written as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%2Ci%27%7D+%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D+1_%7Bc%28h_%2A%29%3Di%2C+c%282h_%2A%29%3Di%27%7D+1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%2Ci%27%7D+%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D+1_%7Bc%28h_%2A%29%3Di%2C+c%282h_%2A%29%3Di%27%7D+1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%2Ci%27%7D+%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D+1_%7Bc%28h_%2A%29%3Di%2C+c%282h_%2A%29%3Di%27%7D+1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i,i'} \sum_{h \in \Phi_j \cap (a_*+H)} 1_{c_0(h)=i} 1_{c(h_*)=i, c(2h_*)=i'} 1_{c(h+h_*)=i'}." class="latex" />
The coupling of the constraints <img src="https://s0.wp.com/latex.php?latex=%7Bc%28h_%2A%29%3Di%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%28h_%2A%29%3Di%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%28h_%2A%29%3Di%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c(h_*)=i}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bc%282h_%2A%29%3Di%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%282h_%2A%29%3Di%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%282h_%2A%29%3Di%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c(2h_*)=i'}" class="latex" /> is annoying (as <img src="https://s0.wp.com/latex.php?latex=%7B%28h_%2A%2C2h_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28h_%2A%2C2h_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28h_%2A%2C2h_%2A%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(h_*,2h_*)}" class="latex" /> is an “infinite complexity” pattern that cannot be controlled by any uniformity norm), but (perhaps surprisingly) will not end up causing an essential difficulty to the argument, as we shall see when we start eliminating the terms in this sum one at a time starting from the right.

We decompose the <img src="https://s0.wp.com/latex.php?latex=%7B1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1_{c(h+h_*)=i'}}" class="latex" /> term using <a href="#chi">(5)</a>: <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D+%3D+P_%7Bi%27%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7D+a%29+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29.+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D+%3D+P_%7Bi%27%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7D+a%29+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29.+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++1_%7Bc%28h%2Bh_%2A%29%3Di%27%7D+%3D+P_%7Bi%27%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7D+a%29+%2B+k_%7Bi%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29+%2B+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29.+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 1_{c(h+h_*)=i'} = P_{i',\varepsilon}(T^{h+h_*+h_M} a) + k_{i,\varepsilon,\varepsilon'}(T^{h+h_*+h_M}a) + g_{i,\varepsilon'}(T^{h+h_*+h_M}a). " class="latex" />
By Markov’s inequality, and removing an exceptional event of probabiilty at most <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa/100}" class="latex" />, we may assume that the <img src="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%27%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%27%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%27%2C%5Cvarepsilon%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g_{i',\varepsilon}}" class="latex" /> have normalized <img src="https://s0.wp.com/latex.php?latex=%7BL%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BL%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BL%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{L^2}" class="latex" /> norm <img src="https://s0.wp.com/latex.php?latex=%7BO_%7B%5Ckappa%2CM_j%7D%28%5Cvarepsilon%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO_%7B%5Ckappa%2CM_j%7D%28%5Cvarepsilon%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO_%7B%5Ckappa%2CM_j%7D%28%5Cvarepsilon%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O_{\kappa,M_j}(\varepsilon)}" class="latex" /> on both of these cosets <img src="https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%2C+2a_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%2C+2a_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba_%2A%2BH%2C+2a_%2A%2BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a_*+H, 2a_*+H}" class="latex" />. As such, the contribution of <img src="https://s0.wp.com/latex.php?latex=%7Bk_%7Bi%27%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk_%7Bi%27%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk_%7Bi%27%2C%5Cvarepsilon%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k_{i',\varepsilon,\varepsilon'}(T^{h+h_*+h_M}a)}" class="latex" /> to <a href="#targ">(6)</a> become negligible if <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon}" class="latex" /> is small enough (depending on <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2Cp%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2Cp%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2Cp%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa,p,M_j}" class="latex" />). From the near weak mixing of the <img src="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%2C%5Cvarepsilon%27%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g_{i,\varepsilon'}}" class="latex" />, we know that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+%7C%5Clangle+T%5Eh+g_%7Bi%2C%5Cvarepsilon%27%7D%2C+g_%7Bi%2C%5Cvarepsilon%27%7D+%5Crangle%7C+%5Cll_%7Bp%2C%5Cvarepsilon%2CM_j%7D+%5Cvarepsilon%27&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+%7C%5Clangle+T%5Eh+g_%7Bi%2C%5Cvarepsilon%27%7D%2C+g_%7Bi%2C%5Cvarepsilon%27%7D+%5Crangle%7C+%5Cll_%7Bp%2C%5Cvarepsilon%2CM_j%7D+%5Cvarepsilon%27&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+%7C%5Clangle+T%5Eh+g_%7Bi%2C%5Cvarepsilon%27%7D%2C+g_%7Bi%2C%5Cvarepsilon%27%7D+%5Crangle%7C+%5Cll_%7Bp%2C%5Cvarepsilon%2CM_j%7D+%5Cvarepsilon%27&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf E}_{h \in \Phi_j \cap (a_*+H)} |\langle T^h g_{i,\varepsilon'}, g_{i,\varepsilon'} \rangle| \ll_{p,\varepsilon,M_j} \varepsilon'" class="latex" />
for all <img src="https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i}" class="latex" />, if we choose <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_j}" class="latex" /> large enough. By genericity of <img src="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Ba%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{a}" class="latex" />, this implies that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+%7C%7B%5Cbf+E%7D_%7Bl+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bl%2Bh_M%7D+a%29+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bl%2Bh_M%7D+a%29%7C+%5Cll_%7Bp%2C%5Cvarepsilon%2CM_j%7D+%5Cvarepsilon%27+%2B+o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+%7C%7B%5Cbf+E%7D_%7Bl+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bl%2Bh_M%7D+a%29+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bl%2Bh_M%7D+a%29%7C+%5Cll_%7Bp%2C%5Cvarepsilon%2CM_j%7D+%5Cvarepsilon%27+%2B+o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+E%7D_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+%7C%7B%5Cbf+E%7D_%7Bl+%5Cin+%7B%5Cbf+F%7D_p%5E%7Bn_M%7D%7D+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bl%2Bh_M%7D+a%29+g_%7Bi%2C%5Cvarepsilon%27%7D%28T%5E%7Bl%2Bh_M%7D+a%29%7C+%5Cll_%7Bp%2C%5Cvarepsilon%2CM_j%7D+%5Cvarepsilon%27+%2B+o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf E}_{h \in \Phi_j \cap (a_*+H)} |{\bf E}_{l \in {\bf F}_p^{n_M}} g_{i,\varepsilon'}(T^{h+l+h_M} a) g_{i,\varepsilon'}(T^{l+h_M} a)| \ll_{p,\varepsilon,M_j} \varepsilon' + o(1)." class="latex" />
From this and standard Cauchy-Schwarz (or van der Corput) arguments we can then show that the contribution of the <img src="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%27%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%27%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg_%7Bi%27%2C%5Cvarepsilon%27%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7Da%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g_{i',\varepsilon'}(T^{h+h_*+h_M}a)}" class="latex" /> to <a href="#targ">(6)</a> is negligible outside of an exceptional event of probability at most <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa/100+o(1)}" class="latex" />, if <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon'}" class="latex" /> is small enough depending on <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2Cp%2CM_j%2C%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2Cp%2CM_j%2C%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2Cp%2CM_j%2C%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa,p,M_j,\varepsilon}" class="latex" />. Finally, the quantity <img src="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%27%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BP_%7Bi%27%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BP_%7Bi%27%2C%5Cvarepsilon%7D%28T%5E%7Bh%2Bh_%2A%2Bh_M%7D+a%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{P_{i',\varepsilon}(T^{h+h_*+h_M} a)}" class="latex" /> is independent of <img src="https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h}" class="latex" />, and in fact is equal up to negligible error to the density of <img src="https://s0.wp.com/latex.php?latex=%7Bc%5E%7B-1%7D%28i%27%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%5E%7B-1%7D%28i%27%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%5E%7B-1%7D%28i%27%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c^{-1}(i')}" class="latex" /> in the coset <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7BM_j%7D%282a_%2A%2BH%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7BM_j%7D%282a_%2A%2BH%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_p%5E%7BM_j%7D%282a_%2A%2BH%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_p^{M_j}(2a_*+H)}" class="latex" />. This density will be <img src="https://s0.wp.com/latex.php?latex=%7B%5Cgg_%7Bp%2C%5Ckappa%2CM_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cgg_%7Bp%2C%5Ckappa%2CM_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cgg_%7Bp%2C%5Ckappa%2CM_j%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\gg_{p,\kappa,M_j}}" class="latex" /> except for those <img src="https://s0.wp.com/latex.php?latex=%7Bi%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i'}" class="latex" /> which would have made a negligible impact on <a href="#targ">(6)</a> in any event due to the rareness of the event <img src="https://s0.wp.com/latex.php?latex=%7Bc%282h_%2A%29%3Di%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%282h_%2A%29%3Di%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%282h_%2A%29%3Di%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c(2h_*)=i'}" class="latex" /> in such cases. As such, to prove <a href="#targ">(6)</a> it suffices to show that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%2Ci%27%7D+%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D+1_%7Bc%28h_%2A%29%3Di%2C+c%282h_%2A%29%3Di%27%7D+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%2Ci%27%7D+%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D+1_%7Bc%28h_%2A%29%3Di%2C+c%282h_%2A%29%3Di%27%7D+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%2Ci%27%7D+%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D+1_%7Bc%28h_%2A%29%3Di%2C+c%282h_%2A%29%3Di%27%7D+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+%281-o%281%29%29+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i,i'} \sum_{h \in \Phi_j \cap (a_*+H)} 1_{c_0(h)=i} 1_{c(h_*)=i, c(2h_*)=i'} \gg_{\kappa,p,M_j} (1-o(1)) |\Phi_j \cap (a_*+H)| " class="latex" />
outside of an event of probability <img src="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Ckappa%2F100%2Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\kappa/100+o(1)}" class="latex" />. Now one can sum in <img src="https://s0.wp.com/latex.php?latex=%7Bi%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i'}" class="latex" /> to simplify the above estiamte to <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%7D+1_%7Bc%28h_%2A%29%3Di%7D+%28%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D%29+%2F+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+1-o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%7D+1_%7Bc%28h_%2A%29%3Di%7D+%28%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D%29+%2F+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+1-o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%7D+1_%7Bc%28h_%2A%29%3Di%7D+%28%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D%29+%2F+%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+1-o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i} 1_{c(h_*)=i} (\sum_{h \in \Phi_j \cap (a_*+H)} 1_{c_0(h)=i}) / |\Phi_j \cap (a_*+H)| \gg_{\kappa,p,M_j} 1-o(1)." class="latex" />
If <img src="https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i}" class="latex" /> is such that <img src="https://s0.wp.com/latex.php?latex=%7B%28%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D%29%2F%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D%29%2F%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28%5Csum_%7Bh+%5Cin+%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7D+1_%7Bc_0%28h%29%3Di%7D%29%2F%7C%5CPhi_j+%5Ccap+%28a_%2A%2BH%29%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(\sum_{h \in \Phi_j \cap (a_*+H)} 1_{c_0(h)=i})/|\Phi_j \cap (a_*+H)|}" class="latex" /> is small compared with <img src="https://s0.wp.com/latex.php?latex=%7Bp%2C%5Ckappa%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bp%2C%5Ckappa%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bp%2C%5Ckappa%2CM_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{p,\kappa,M_j}" class="latex" />, then by genericity (and assuming <img src="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5CPhi_j%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Phi_j}" class="latex" /> large enough), the probability that <img src="https://s0.wp.com/latex.php?latex=%7Bc%28h_%2A%29%3Di%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bc%28h_%2A%29%3Di%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bc%28h_%2A%29%3Di%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{c(h_*)=i}" class="latex" /> will similarly be small (up to <img src="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bo%281%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{o(1)}" class="latex" /> errors), and thus have a negligible influence on the above sum. As such, the above estimate simplifies to <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%7D+1_%7Bc%28h_%2A%29%3Di%7D+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+1-o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%7D+1_%7Bc%28h_%2A%29%3Di%7D+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+1-o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bi%7D+1_%7Bc%28h_%2A%29%3Di%7D+%5Cgg_%7B%5Ckappa%2Cp%2CM_j%7D+1-o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i} 1_{c(h_*)=i} \gg_{\kappa,p,M_j} 1-o(1)." class="latex" />
But the left-hand side sums to one, and the claim follows.

]]></content:encoded>
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<media:title type="html">Terry</media:title>
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<item>
<title>Two announcements: AI for Math resources, and erdosproblems.com</title>
<link>https://terrytao.wordpress.com/2024/04/19/two-announcements-ai-for-math-resources-and-erdosproblems-com/</link>
<comments>https://terrytao.wordpress.com/2024/04/19/two-announcements-ai-for-math-resources-and-erdosproblems-com/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Fri, 19 Apr 2024 20:47:40 +0000</pubDate>
<category><![CDATA[advertising]]></category>
<category><![CDATA[Mathematics]]></category>
<category><![CDATA[question]]></category>
<category><![CDATA[Artificial Intelligence]]></category>
<category><![CDATA[Paul Erdos]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14455</guid>
<description><![CDATA[This post contains two unrelated announcements. Firstly, I would like to promote a useful list of resources for AI in Mathematics, that was initiated by Talia Ringer (with the crowdsourced assistance of many others) during the National Academies workshop on “AI in mathematical reasoning” last year. This list is now accepting new contributions, updates, or […]]]></description>
<content:encoded><![CDATA[
This post contains two unrelated announcements. Firstly, I would like to promote a useful <a href="https://docs.google.com/document/d/1kD7H4E28656ua8jOGZ934nbH2HcBLyxcRgFDduH5iQ0/edit">list of resources for AI in Mathematics</a>, that was initiated by <a href="https://dependenttyp.es/">Talia Ringer</a> (with the crowdsourced assistance of many others) during the <a href="https://www.nationalacademies.org/our-work/ai-to-assist-mathematical-reasoning-a-workshop">National Academies workshop on “AI in mathematical reasoning”</a> last year. This list is now accepting new contributions, updates, or corrections; please feel free to submit them <a href="https://docs.google.com/document/d/1kD7H4E28656ua8jOGZ934nbH2HcBLyxcRgFDduH5iQ0/edit">directly to the list</a> (which I am helping Talia to edit). Incidentally, next week there will be a second <a href="https://www.nationalacademies.org/event/42508_04-2024_artificial-intelligence-to-assist-mathematical-reasoning-webinar-on-the-future-of-collaboration">followup webinar</a> to the aforementioned workshop, building on the topics covered there. (The first webinar may be found <a href="https://www.nationalacademies.org/event/42508_04-2024_artificial-intelligence-to-assist-mathematical-reasoning-webinar-on-the-future-of-collaboration">here</a>.)
Secondly, I would like to advertise the <a href="https://www.erdosproblems.com/">erdosproblems.com</a> website, launched recently by <a href="https://www.thomasbloom.org/">Thomas Bloom</a>. This is intended to be a living repository of the many mathematical problems proposed in various venues by <a href="https://en.wikipedia.org/wiki/Paul_Erd%C5%91s">Paul Erdős</a>, who was particularly noted for his influential posing of such problems. For a tour of the site and an explanation of its purpose, I can recommend Thomas’s <a href="https://www.youtube.com/watch?v=snAjIjo8Qy4">recent talk on this topic</a> at a <a href="https://www.newton.ac.uk/event/ooew04/">conference last week in honor of Timothy Gowers</a>.
Thomas is currently issuing a <a href="https://www.erdosproblems.com/help">call for help</a> to develop the erdosproblems.com website in a number of ways (quoting directly from that page):
<ul>
<li>You know Github and could set a suitable project up to allow people to contribute new problems (and corrections to old ones) to the database, and could help me maintain the Github project;</li>
<li>You know things about web design and have suggestions for how this website could look or perform better;</li>
<li>You know things about Python/Flask/HTML/SQL/whatever and want to help me code cool new features on the website;</li>
<li>You know about accessibility and have an idea how I can make this website more accessible (to any group of people);</li>
<li>You are a mathematician who has thought about some of the problems here and wants to write an expanded commentary for one of them, with lots of references, comparisons to other problems, and other miscellaneous insights (mathematician here is interpreted broadly, in that if you have thought about the problems on this site and are willing to write such a commentary you qualify);</li>
<li>You knew Erdős and have any memories or personal correspondence concerning a particular problem;</li>
<li>You have solved an Erdős problem and I’ll update the website accordingly (and apologies if you solved this problem some time ago);</li>
<li>You have spotted a mistake, typo, or duplicate problem, or anything else that has confused you and I’ll correct things;</li>
<li>You are a human being with an internet connection and want to volunteer a particular Erdős paper or problem list to go through and add new problems from (please let me know before you start, to avoid duplicate efforts);</li>
<li>You have any other ideas or suggestions – there are probably lots of things I haven’t thought of, both in ways this site can be made better, and also what else could be done from this project. Please get in touch with any ideas!</li>
</ul>
I for instance contributed a <a href="https://www.erdosproblems.com/587">problem to the site</a> (#587) that Erdős himself gave to me personally (this was the topic of a somewhat well known photo of Paul and myself, and which he communicated again to be shortly afterwards on a postcard; links to both images can be found by following the above link). As it turns out, this particular problem was essentially solved in 2010 <a href="https://zbmath.org/1222.11012">by Nguyen and Vu</a>.
(Incidentally, I also spoke at the same conference that Thomas spoke at, on <a href="https://terrytao.wordpress.com/2024/04/04/martons-conjecture-in-abelian-groups-with-bounded-torsion/">my recent work with Gowers, Green, and Manners</a>; <a href="https://www.youtube.com/watch?v=F35rUbHTWDo">here is the video of my talk</a>, and <a href="https://terrytao.files.wordpress.com/2024/04/pfr.pdf">here are my slides</a>.)
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<media:title type="html">Terry</media:title>
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<title>Marton’s conjecture in abelian groups with bounded torsion</title>
<link>https://terrytao.wordpress.com/2024/04/04/martons-conjecture-in-abelian-groups-with-bounded-torsion/</link>
<comments>https://terrytao.wordpress.com/2024/04/04/martons-conjecture-in-abelian-groups-with-bounded-torsion/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Thu, 04 Apr 2024 22:05:52 +0000</pubDate>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[paper]]></category>
<category><![CDATA[additive combinatorics]]></category>
<category><![CDATA[Ben Green]]></category>
<category><![CDATA[Freddie Manners]]></category>
<category><![CDATA[Polynomial Freiman-Ruzsa conjecture]]></category>
<category><![CDATA[Shannon entropy]]></category>
<category><![CDATA[Timothy Gowers]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14442</guid>
<description><![CDATA[Tim Gowers, Ben Green, Freddie Manners, and I have just uploaded to the arXiv our paper “Marton’s conjecture in abelian groups with bounded torsion“. This paper fully resolves a conjecture of Katalin Marton (the bounded torsion case of the Polynomial Freiman–Ruzsa conjecture (first proposed by Katalin Marton): Theorem 1 (Marton’s conjecture) Let be an abelian […]]]></description>
<content:encoded><![CDATA[<a href="https://www.dpmms.cam.ac.uk/~wtg10/">Tim Gowers</a>, <a href="https://people.maths.ox.ac.uk/greenbj/">Ben Green</a>, <a href="https://mathweb.ucsd.edu/~fmanners/">Freddie Manners</a>, and I have just uploaded to the arXiv our paper “<a href="https://arxiv.org/abs/2404.02244">Marton’s conjecture in abelian groups with bounded torsion</a>“. This paper fully resolves a conjecture of Katalin Marton (the bounded torsion case of the <a href="https://terrytao.wordpress.com/2007/03/11/ben-green-the-polynomial-freiman-ruzsa-conjecture/">Polynomial Freiman–Ruzsa conjecture</a> (first proposed by Katalin Marton):
<blockquote>Theorem 1 (Marton’s conjecture) <a name="main"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG+%3D+%28G%2C%2B%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G = (G,+)}" class="latex" /> be an abelian <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-torsion group (thus, <img src="https://s0.wp.com/latex.php?latex=%7Bmx%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bmx%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bmx%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{mx=0}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=%7Bx+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%5Cin+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x \in G}" class="latex" />), and let <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+G%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \subset G}" class="latex" /> be such that <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A+A| \leq K|A|}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> can be covered by at most <img src="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7BO%28m%5E3%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7BO%28m%5E3%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7BO%28m%5E3%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(2K)^{O(m^3)}}" class="latex" /> translates of a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> of cardinality at most <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A|}" class="latex" />. Moreover, <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> is contained in <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell A - \ell A}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+m+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+m+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+m+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell \ll (2 + m \log K)^{O(m^3 \log m)}}" class="latex" />.</blockquote>
We had <a href="https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/">previously established</a> the <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> case of this result, with the number of translates bounded by <img src="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(2K)^{12}}" class="latex" /> (which was subsequently improved to <img src="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B11%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B11%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%282K%29%5E%7B11%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(2K)^{11}}" class="latex" /> by Jyun-Jie Liao), but without the additional containment <img src="https://s0.wp.com/latex.php?latex=%7BH+%5Csubset+%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH+%5Csubset+%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH+%5Csubset+%5Cell+A+-+%5Cell+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H \subset \ell A - \ell A}" class="latex" />. It remains a challenge to replace <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell}" class="latex" /> by a bounded constant (such as <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />); this is essentially the “polynomial Bogolyubov conjecture”, which is still open. The <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> result has been formalized in the proof assistant language Lean, as discussed in <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">this previous blog post</a>. As a consequence of this result, many of the applications of the previous theorem may now be extended from characteristic <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" /> to higher characteristic.
Our proof techniques are a modification of those in our previous paper, and in particular continue to be based on the theory of Shannon entropy. For inductive purposes, it turns out to be convenient to work with the following version of the conjecture (which, up to <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-dependent constants, is actually equivalent to the above theorem):
<blockquote>Theorem 2 (Marton’s conjecture, entropy form) <a name="main-entropy"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> be an abelian <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-torsion group, and let <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,\dots,X_m}" class="latex" /> be independent finitely supported random variables on <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" />, such that
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+%5Cleq+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+%5Cleq+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+%5Cleq+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_1+\dots+X_m] - \frac{1}{m} \sum_{i=1}^m {\bf H}[X_i] \leq \log K," class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+H%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+H%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+H%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf H}}" class="latex" /> denotes Shannon entropy. Then there is a uniform random variable <img src="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U_H}" class="latex" /> on a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> such that
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+d%5BX_i%3B+U_H%5D+%5Cll+m%5E3+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+d%5BX_i%3B+U_H%5D+%5Cll+m%5E3+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+d%5BX_i%3B+U_H%5D+%5Cll+m%5E3+%5Clog+K%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \frac{1}{m} \sum_{i=1}^m d[X_i; U_H] \ll m^3 \log K," class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d}" class="latex" /> denotes the entropic Ruzsa distance (see <a href="https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/">previous blog post</a> for a definition); furthermore, if all the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> take values in some symmetric set <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />, then <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> lies in <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+S%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+S%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+S%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell S}" class="latex" /> for some <img src="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cell+%5Cll+%282+%2B+%5Clog+K%29%5E%7BO%28m%5E3+%5Clog+m%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\ell \ll (2 + \log K)^{O(m^3 \log m)}}" class="latex" />.</blockquote>
As a first approximation, one should think of all the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> as identically distributed, and having the uniform distribution on <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" />, as this is the case that is actually relevant for implying Theorem <a href="#main">1</a>; however, the recursive nature of the proof of Theorem <a href="#main-entropy">2</a> requires one to manipulate the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> separately. It also is technically convenient to work with <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" /> independent variables, rather than just a pair of variables as we did in the <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> case; this is perhaps the biggest additional technical complication needed to handle higher characteristics.
The strategy, as with the previous paper, is to attempt an entropy decrement argument: to try to locate modifications <img src="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X'_1,\dots,X'_m}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2C%5Cdots%2CX_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,\dots,X_m}" class="latex" /> that are reasonably close (in Ruzsa distance) to the original random variables, while decrementing the “multidistance”
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%5Cbf+H%7D%5BX_1%2B%5Cdots%2BX_m%5D+-+%5Cfrac%7B1%7D%7Bm%7D+%5Csum_%7Bi%3D1%7D%5Em+%7B%5Cbf+H%7D%5BX_i%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_1+\dots+X_m] - \frac{1}{m} \sum_{i=1}^m {\bf H}[X_i] " class="latex" />
which turns out to be a convenient metric for progress (for instance, this quantity is non-negative, and vanishes if and only if the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> are all translates of a uniform random variable <img src="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU_H%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U_H}" class="latex" /> on a subgroup <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" />). In the previous paper we modified the corresponding functional to minimize by some additional terms in order to improve the exponent <img src="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{12}" class="latex" />, but as we are not attempting to completely optimize the constants, we did not do so in the current paper (and as such, our arguments here give a slightly different way of establishing the <img src="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m=2}" class="latex" /> case, albeit with somewhat worse exponents).
As before, we search for such improved random variables <img src="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%27_1%2C%5Cdots%2CX%27_m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X'_1,\dots,X'_m}" class="latex" /> by introducing more independent random variables – we end up taking an array of <img src="https://s0.wp.com/latex.php?latex=%7Bm%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m^2}" class="latex" /> random variables <img src="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y_{i,j}}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7Bi%2Cj%3D1%2C%5Cdots%2Cm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%2Cj%3D1%2C%5Cdots%2Cm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%2Cj%3D1%2C%5Cdots%2Cm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i,j=1,\dots,m}" class="latex" />, with each <img src="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y_{i,j}}" class="latex" /> a copy of <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" />, and forming various sums of these variables and conditioning them against other sums. Thanks to the magic of Shannon entropy inequalities, it turns out that it is guaranteed that at least one of these modifications will decrease the multidistance, except in an “endgame” situation in which certain random variables are nearly (conditionally) independent of each other, in the sense that certain conditional mutual informations are small. In particular, in the endgame scenario, the row sums <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_j+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_j+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_j+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_j Y_{i,j}}" class="latex" /> of our array will end up being close to independent of the column sums <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_i+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_i+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_i+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_i Y_{i,j}}" class="latex" />, subject to conditioning on the total sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{i,j} Y_{i,j}}" class="latex" />. Not coincidentally, this type of conditional independence phenomenon also shows up when considering row and column sums of iid independent gaussian random variables, as a specific feature of the gaussian distribution. It is related to the more familiar observation that if <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y}" class="latex" /> are two independent copies of a Gaussian random variable, then <img src="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X+Y}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX-Y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX-Y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX-Y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X-Y}" class="latex" /> are also independent of each other.
Up until now, the argument does not use the <img src="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m}" class="latex" />-torsion hypothesis, nor the fact that we work with an <img src="https://s0.wp.com/latex.php?latex=%7Bm+%5Ctimes+m%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bm+%5Ctimes+m%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bm+%5Ctimes+m%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{m \times m}" class="latex" /> array of random variables as opposed to some other shape of array. But now the torsion enters in a key role, via the obvious identity
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%2Cj%7D+i+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+j+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+%28-i-j%29+Y_%7Bi%2Cj%7D+%3D+0.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%2Cj%7D+i+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+j+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+%28-i-j%29+Y_%7Bi%2Cj%7D+%3D+0.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%2Cj%7D+i+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+j+Y_%7Bi%2Cj%7D+%2B+%5Csum_%7Bi%2Cj%7D+%28-i-j%29+Y_%7Bi%2Cj%7D+%3D+0.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i,j} i Y_{i,j} + \sum_{i,j} j Y_{i,j} + \sum_{i,j} (-i-j) Y_{i,j} = 0." class="latex" />
In the endgame, the any pair of these three random variables are close to independent (after conditioning on the total sum <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bi%2Cj%7D+Y_%7Bi%2Cj%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{i,j} Y_{i,j}}" class="latex" />). Applying some “entropic Ruzsa calculus” (and in particular an entropic version of the Balog–Szeméredi–Gowers inequality), one can then arrive at a new random variable <img src="https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U}" class="latex" /> of small entropic doubling that is reasonably close to all of the <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> in Ruzsa distance, which provides the final way to reduce the multidistance.
Besides the polynomial Bogolyubov conjecture mentioned above (which we do not know how to address by entropy methods), the other natural question is to try to develop a characteristic zero version of this theory in order to establish the polynomial Freiman–Ruzsa conjecture over torsion-free groups, which in our language asserts (roughly speaking) that random variables of small entropic doubling are close (in Ruzsa distance) to a discrete Gaussian random variable, with good bounds. The above machinery is consistent with this conjecture, in that it produces lots of independent variables related to the original variable, various linear combinations of which obey the same sort of entropy estimates that gaussian random variables would exhibit, but what we are missing is a way to get back from these entropy estimates to an assertion that the random variables really are close to Gaussian in some sense. In continuous settings, Gaussians are known to extremize the entropy for a given variance, and of course we have the central limit theorem that shows that averages of random variables typically converge to a Gaussian, but it is not clear how to adapt these phenomena to the discrete Gaussian setting (without the circular reasoning of assuming the polynoimal Freiman–Ruzsa conjecture to begin with).
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<media:title type="html">Terry</media:title>
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<item>
<title>AI Mathematical Olympiad – Progress Prize Competition now open</title>
<link>https://terrytao.wordpress.com/2024/04/02/ai-mathematical-olympiad-progress-prize-competition-now-open/</link>
<comments>https://terrytao.wordpress.com/2024/04/02/ai-mathematical-olympiad-progress-prize-competition-now-open/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Tue, 02 Apr 2024 23:34:55 +0000</pubDate>
<category><![CDATA[advertising]]></category>
<category><![CDATA[math.GM]]></category>
<category><![CDATA[Artificial Intelligence]]></category>
<category><![CDATA[international mathematical olympiad]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14434</guid>
<description><![CDATA[The first progress prize competition for the AI Mathematical Olympiad has now launched. (Disclosure: I am on the advisory committee for the prize.) This is a competition in which contestants submit an AI model which, after the submissions deadline on June 27, will be tested (on a fixed computational resource, without internet access) on a […]]]></description>
<content:encoded><![CDATA[
The first progress prize competition for the <a href="https://aimoprize.com/">AI Mathematical Olympiad</a> has <a href="https://www.kaggle.com/competitions/ai-mathematical-olympiad-prize/overview">now launched</a>. (Disclosure: I am on the <a href="https://aimoprize.com/updates/2024-02-07-advisory-committee-announced">advisory committee</a> for the prize.) This is a competition in which contestants submit an AI model which, after the submissions deadline on June 27, will be tested (on a fixed computational resource, without internet access) on a set of 50 “private” test math problems, each of which has an answer as an integer between 0 and 999. Prior to the close of submission, the models can be tested on 50 “public” test math problems (where the results of the model are public, but not the problems themselves), as well as 10 training problems that are available to all contestants. As of this time of writing, the <a href="https://www.kaggle.com/competitions/ai-mathematical-olympiad-prize/leaderboard">leaderboard </a>shows that the best-performing model has solved 4 out of 50 of the questions (a standard benchmark, Gemma 7B, had previously solved 3 out of 50). A total of $<img src="https://s0.wp.com/latex.php?latex=2%5E%7B20%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2%5E%7B20%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2%5E%7B20%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2^{20}" class="latex" /> ($1.048 million) has been allocated for various prizes associated to this competition. More detailed rules can be found <a href="https://www.kaggle.com/competitions/ai-mathematical-olympiad-prize/rules">here</a>.


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<slash:comments>25</slash:comments>
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<media:title type="html">Terry</media:title>
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<item>
<title>Talks at the JMM</title>
<link>https://terrytao.wordpress.com/2024/03/17/talks-at-the-jmm/</link>
<comments>https://terrytao.wordpress.com/2024/03/17/talks-at-the-jmm/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Mon, 18 Mar 2024 05:07:28 +0000</pubDate>
<category><![CDATA[advertising]]></category>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[math.HO]]></category>
<category><![CDATA[math.NT]]></category>
<category><![CDATA[talk]]></category>
<category><![CDATA[Joint mathematics meetings]]></category>
<category><![CDATA[machine assisted proof]]></category>
<category><![CDATA[multiplicative functions]]></category>
<category><![CDATA[periodic tiling conjecture]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14419</guid>
<description><![CDATA[Earlier this year, I gave a series of lectures at the Joint Mathematics Meetings at San Francisco. I am uploading here the slides for these talks: I also have written a text version of the first talk, which has been submitted to the Notices of the American Mathematical Society.]]></description>
<content:encoded><![CDATA[
Earlier this year, I gave a series of lectures at the <a href="https://www.jointmathematicsmeetings.org/meetings/national/jmm2024/2300_program.html">Joint Mathematics Meetings at San Francisco</a>. I am uploading here the slides for these talks:
<ul>
<li>“<a href="https://terrytao.files.wordpress.com/2024/03/machine-jan-3.pdf">Machine assisted proof</a>” (<a href="https://www.youtube.com/watch?v=AayZuuDDKP0">Video here</a>)</li>
<li>“<a href="https://terrytao.files.wordpress.com/2024/03/periodic-tiling-jan-4.pdf">Translational tilings of Euclidean space</a>” (<a href="https://www.youtube.com/watch?v=QQBqJZtWIhA">Video here</a>)</li>
<li>“<a href="https://terrytao.files.wordpress.com/2024/03/correlations-jan-5.pdf">Correlations of multiplicative functions</a>” (<a href="https://www.youtube.com/watch?v=t_plilnbAtM">Video here</a>)</li>
</ul>
I also have written a <a href="https://terrytao.files.wordpress.com/2024/03/machine-assisted-proof-notices.pdf">text version of the first talk</a>, which has been submitted to the Notices of the American Mathematical Society.
]]></content:encoded>
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<media:title type="html">Terry</media:title>
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<item>
<title>A generalized Cauchy-Schwarz inequality via the Gibbs variational formula</title>
<link>https://terrytao.wordpress.com/2023/12/10/a-generalized-cauchy-schwarz-inequality-via-the-gibbs-variational-formula/</link>
<comments>https://terrytao.wordpress.com/2023/12/10/a-generalized-cauchy-schwarz-inequality-via-the-gibbs-variational-formula/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Mon, 11 Dec 2023 03:23:52 +0000</pubDate>
<category><![CDATA[expository]]></category>
<category><![CDATA[math.CA]]></category>
<category><![CDATA[math.PR]]></category>
<category><![CDATA[Anthony Carbery]]></category>
<category><![CDATA[Cauchy-Schwarz]]></category>
<category><![CDATA[Gibbs variational formula]]></category>
<category><![CDATA[Shannon entropy]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14307</guid>
<description><![CDATA[Let be a non-empty finite set. If is a random variable taking values in , the Shannon entropy of is defined as There is a nice variational formula that lets one compute logs of sums of exponentials in terms of this entropy: Lemma 1 (Gibbs variational formula) Let be a function. Then Proof: Note that […]]]></description>
<content:encoded><![CDATA[

Let <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" /> be a non-empty finite set. If <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is a random variable taking values in <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />, the Shannon entropy <img src="https://s0.wp.com/latex.php?latex=%7BH%5BX%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%5BX%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%5BX%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H[X]}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is defined as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+H%5BX%5D+%3D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+H%5BX%5D+%3D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+H%5BX%5D+%3D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle H[X] = -\sum_{s \in S} {\bf P}[X = s] \log {\bf P}[X = s]." class="latex" />
There is a nice variational formula that lets one compute logs of sums of exponentials in terms of this entropy:

<blockquote>Lemma 1 (Gibbs variational formula) <a name="gibbs"></a> Let <img src="https://s0.wp.com/latex.php?latex=%7Bf%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f: S \rightarrow {\bf R}}" class="latex" /> be a function. Then <a name="efs"><img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+%5Csup_X+%7B%5Cbf+E%7D+f%28X%29+%2B+%7B%5Cbf+H%7D%5BX%5D.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+%5Csup_X+%7B%5Cbf+E%7D+f%28X%29+%2B+%7B%5Cbf+H%7D%5BX%5D.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+%5Csup_X+%7B%5Cbf+E%7D+f%28X%29+%2B+%7B%5Cbf+H%7D%5BX%5D.+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \log \sum_{s \in S} \exp(f(s)) = \sup_X {\bf E} f(X) + {\bf H}[X]. \ \ \ \ \ (1)" class="latex" />
</a> </blockquote>


Proof: Note that shifting <img src="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f}" class="latex" /> by a constant affects both sides of <a href="#efs">(1)</a> the same way, so we may normalize <img src="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csum_%7Bs+%5Cin+S%7D+%5Cexp%28f%28s%29%29+%3D+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sum_{s \in S} \exp(f(s)) = 1}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28f%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28f%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cexp%28f%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\exp(f(s))}" class="latex" /> is now the probability distribution of some random variable <img src="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y}" class="latex" />, and the inequality can be rewritten as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++0+%3D+%5Csup_X+%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BY+%3D+s%5D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++0+%3D+%5Csup_X+%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BY+%3D+s%5D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++0+%3D+%5Csup_X+%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BY+%3D+s%5D+-%5Csum_%7Bs+%5Cin+S%7D+%7B%5Cbf+P%7D%5BX+%3D+s%5D+%5Clog+%7B%5Cbf+P%7D%5BX+%3D+s%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 0 = \sup_X \sum_{s \in S} {\bf P}[X = s] \log {\bf P}[Y = s] -\sum_{s \in S} {\bf P}[X = s] \log {\bf P}[X = s]." class="latex" />
But this is precisely the <a href="https://en.wikipedia.org/wiki/Gibbs%27_inequality">Gibbs inequality</a>. (The expression inside the supremum can also be written as <img src="https://s0.wp.com/latex.php?latex=%7B-D_%7BKL%7D%28X%7C%7CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B-D_%7BKL%7D%28X%7C%7CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B-D_%7BKL%7D%28X%7C%7CY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{-D_{KL}(X||Y)}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7BD_%7BKL%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BD_%7BKL%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BD_%7BKL%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{D_{KL}}" class="latex" /> denotes <a href="https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence">Kullback-Leibler divergence</a>. One can also interpret this inequality as a special case of <a href="https://en.wikipedia.org/wiki/Convex_conjugate#Fenchel's_inequality">the Fenchel–Young inequality</a> relating the conjugate convex functions <img src="https://s0.wp.com/latex.php?latex=%7Bx+%5Cmapsto+e%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bx+%5Cmapsto+e%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bx+%5Cmapsto+e%5Ex%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{x \mapsto e^x}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7By+%5Cmapsto+y+%5Clog+y+-+y%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7By+%5Cmapsto+y+%5Clog+y+-+y%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7By+%5Cmapsto+y+%5Clog+y+-+y%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{y \mapsto y \log y - y}" class="latex" />.) <img src="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\Box" class="latex" />

In this note I would like to use this variational formula (which is also known as the Donsker-Varadhan variational formula) to give another proof of the following <a href="https://zbmath.org/1043.05011">inequality of Carbery</a>.

<blockquote>Theorem 2 (Generalized Cauchy-Schwarz inequality) Let <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \geq 0}" class="latex" />, let <img src="https://s0.wp.com/latex.php?latex=%7BS%2C+T_1%2C%5Cdots%2CT_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%2C+T_1%2C%5Cdots%2CT_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%2C+T_1%2C%5Cdots%2CT_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S, T_1,\dots,T_n}" class="latex" /> be finite non-empty sets, and let <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%3A+S+%5Crightarrow+T_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%3A+S+%5Crightarrow+T_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%3A+S+%5Crightarrow+T_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_i: S \rightarrow T_i}" class="latex" /> be functions for each <img src="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i=1,\dots,n}" class="latex" />. Let <img src="https://s0.wp.com/latex.php?latex=%7BK%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K: S \rightarrow {\bf R}^+}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%5E%2B%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i: T_i \rightarrow {\bf R}^+}" class="latex" /> be positive functions for each <img src="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i=1,\dots,n}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+%5Cprod_%7Bi%3D1%7D%5En+f_i%28%5Cpi_i%28s%29%29+%5Cleq+Q+%5Cprod_%7Bi%3D1%7D%5En+%28%5Csum_%7Bt_i+%5Cin+T_i%7D+f_i%28t_i%29%5E%7Bn%2B1%7D%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+%5Cprod_%7Bi%3D1%7D%5En+f_i%28%5Cpi_i%28s%29%29+%5Cleq+Q+%5Cprod_%7Bi%3D1%7D%5En+%28%5Csum_%7Bt_i+%5Cin+T_i%7D+f_i%28t_i%29%5E%7Bn%2B1%7D%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+%5Cprod_%7Bi%3D1%7D%5En+f_i%28%5Cpi_i%28s%29%29+%5Cleq+Q+%5Cprod_%7Bi%3D1%7D%5En+%28%5Csum_%7Bt_i+%5Cin+T_i%7D+f_i%28t_i%29%5E%7Bn%2B1%7D%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{s \in S} K(s) \prod_{i=1}^n f_i(\pi_i(s)) \leq Q \prod_{i=1}^n (\sum_{t_i \in T_i} f_i(t_i)^{n+1})^{1/(n+1)}" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BQ%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Q}" class="latex" /> is the quantity <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Q+%3A%3D+%28%5Csum_%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+%5COmega_n%7D+K%28s_0%29+%5Cdots+K%28s_n%29%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Q+%3A%3D+%28%5Csum_%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+%5COmega_n%7D+K%28s_0%29+%5Cdots+K%28s_n%29%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++Q+%3A%3D+%28%5Csum_%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+%5COmega_n%7D+K%28s_0%29+%5Cdots+K%28s_n%29%29%5E%7B1%2F%28n%2B1%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle Q := (\sum_{(s_0,\dots,s_n) \in \Omega_n} K(s_0) \dots K(s_n))^{1/(n+1)}" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" /> is the set of all tuples <img src="https://s0.wp.com/latex.php?latex=%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+S%5E%7Bn%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+S%5E%7Bn%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28s_0%2C%5Cdots%2Cs_n%29+%5Cin+S%5E%7Bn%2B1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(s_0,\dots,s_n) \in S^{n+1}}" class="latex" /> such that <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%28s_%7Bi-1%7D%29+%3D+%5Cpi_i%28s_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%28s_%7Bi-1%7D%29+%3D+%5Cpi_i%28s_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_i%28s_%7Bi-1%7D%29+%3D+%5Cpi_i%28s_i%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_i(s_{i-1}) = \pi_i(s_i)}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bi%3D1%2C%5Cdots%2Cn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{i=1,\dots,n}" class="latex" />. </blockquote>


Thus for instance, the identity is trivial for <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=0}" class="latex" />. When <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=1}" class="latex" />, the inequality reads <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+%5Cleq+%28%5Csum_%7Bs_0%2Cs_1+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%7D+K%28s_0%29+K%28s_1%29%29%5E%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+%5Cleq+%28%5Csum_%7Bs_0%2Cs_1+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%7D+K%28s_0%29+K%28s_1%29%29%5E%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+%5Cleq+%28%5Csum_%7Bs_0%2Cs_1+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%7D+K%28s_0%29+K%28s_1%29%29%5E%7B1%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{s \in S} K(s) f_1(\pi_1(s)) \leq (\sum_{s_0,s_1 \in S: \pi_1(s_0)=\pi_1(s_1)} K(s_0) K(s_1))^{1/2}" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28+%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E2%29%5E%7B1%2F2%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28+%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E2%29%5E%7B1%2F2%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28+%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E2%29%5E%7B1%2F2%7D%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle ( \sum_{t_1 \in T_1} f_1(t_1)^2)^{1/2}," class="latex" />
which is easily proven by Cauchy-Schwarz, while for <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=2}" class="latex" /> the inequality reads <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+f_2%28%5Cpi_2%28s%29%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+f_2%28%5Cpi_2%28s%29%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bs+%5Cin+S%7D+K%28s%29+f_1%28%5Cpi_1%28s%29%29+f_2%28%5Cpi_2%28s%29%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{s \in S} K(s) f_1(\pi_1(s)) f_2(\pi_2(s)) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%28%5Csum_%7Bs_0%2Cs_1%2C+s_2+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%3B+%5Cpi_2%28s_1%29%3D%5Cpi_2%28s_2%29%7D+K%28s_0%29+K%28s_1%29+K%28s_2%29%29%5E%7B1%2F3%7D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%28%5Csum_%7Bs_0%2Cs_1%2C+s_2+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%3B+%5Cpi_2%28s_1%29%3D%5Cpi_2%28s_2%29%7D+K%28s_0%29+K%28s_1%29+K%28s_2%29%29%5E%7B1%2F3%7D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%28%5Csum_%7Bs_0%2Cs_1%2C+s_2+%5Cin+S%3A+%5Cpi_1%28s_0%29%3D%5Cpi_1%28s_1%29%3B+%5Cpi_2%28s_1%29%3D%5Cpi_2%28s_2%29%7D+K%28s_0%29+K%28s_1%29+K%28s_2%29%29%5E%7B1%2F3%7D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \leq (\sum_{s_0,s_1, s_2 \in S: \pi_1(s_0)=\pi_1(s_1); \pi_2(s_1)=\pi_2(s_2)} K(s_0) K(s_1) K(s_2))^{1/3} " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E3%29%5E%7B1%2F3%7D+%28%5Csum_%7Bt_2+%5Cin+T_2%7D+f_2%28t_2%29%5E3%29%5E%7B1%2F3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E3%29%5E%7B1%2F3%7D+%28%5Csum_%7Bt_2+%5Cin+T_2%7D+f_2%28t_2%29%5E3%29%5E%7B1%2F3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28%5Csum_%7Bt_1+%5Cin+T_1%7D+f_1%28t_1%29%5E3%29%5E%7B1%2F3%7D+%28%5Csum_%7Bt_2+%5Cin+T_2%7D+f_2%28t_2%29%5E3%29%5E%7B1%2F3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle (\sum_{t_1 \in T_1} f_1(t_1)^3)^{1/3} (\sum_{t_2 \in T_2} f_2(t_2)^3)^{1/3}" class="latex" />
which can also be proven by elementary means. However even for <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=3}" class="latex" />, the existing proofs require the “tensor power trick” in order to reduce to the case when the <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i}" class="latex" /> are step functions (in which case the inequality can be proven elementarily, as discussed in the above paper of Carbery).

We now prove this inequality. We write <img src="https://s0.wp.com/latex.php?latex=%7BK%28s%29+%3D+%5Cexp%28k%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%28s%29+%3D+%5Cexp%28k%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%28s%29+%3D+%5Cexp%28k%28s%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K(s) = \exp(k(s))}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bf_i%28t_i%29+%3D+%5Cexp%28g_i%28t_i%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bf_i%28t_i%29+%3D+%5Cexp%28g_i%28t_i%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bf_i%28t_i%29+%3D+%5Cexp%28g_i%28t_i%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{f_i(t_i) = \exp(g_i(t_i))}" class="latex" /> for some functions <img src="https://s0.wp.com/latex.php?latex=%7Bk%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bk%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bk%3A+S+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{k: S \rightarrow {\bf R}}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7Bg_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bg_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bg_i%3A+T_i+%5Crightarrow+%7B%5Cbf+R%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{g_i: T_i \rightarrow {\bf R}}" class="latex" />. If we take logarithms in the inequality to be proven and apply Lemma <a href="#gibbs">1</a>, the inequality becomes <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_X+%7B%5Cbf+E%7D+k%28X%29+%2B+%5Csum_%7Bi%3D1%7D%5En+g_i%28%5Cpi_i%28X%29%29+%2B+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_X+%7B%5Cbf+E%7D+k%28X%29+%2B+%5Csum_%7Bi%3D1%7D%5En+g_i%28%5Cpi_i%28X%29%29+%2B+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csup_X+%7B%5Cbf+E%7D+k%28X%29+%2B+%5Csum_%7Bi%3D1%7D%5En+g_i%28%5Cpi_i%28X%29%29+%2B+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sup_X {\bf E} k(X) + \sum_{i=1}^n g_i(\pi_i(X)) + {\bf H}[X] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csup_%7B%28X_0%2C%5Cdots%2CX_n%29%7D+%7B%5Cbf+E%7D+k%28X_0%29%2B%5Cdots%2Bk%28X_n%29+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csup_%7B%28X_0%2C%5Cdots%2CX_n%29%7D+%7B%5Cbf+E%7D+k%28X_0%29%2B%5Cdots%2Bk%28X_n%29+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cleq+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csup_%7B%28X_0%2C%5Cdots%2CX_n%29%7D+%7B%5Cbf+E%7D+k%28X_0%29%2B%5Cdots%2Bk%28X_n%29+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \leq \frac{1}{n+1} \sup_{(X_0,\dots,X_n)} {\bf E} k(X_0)+\dots+k(X_n) + {\bf H}[X_0,\dots,X_n] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csum_%7Bi%3D1%7D%5En+%5Csup_%7BY_i%7D+%28n%2B1%29+%7B%5Cbf+E%7D+g_i%28Y_i%29+%2B+%7B%5Cbf+H%7D%5BY_i%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csum_%7Bi%3D1%7D%5En+%5Csup_%7BY_i%7D+%28n%2B1%29+%7B%5Cbf+E%7D+g_i%28Y_i%29+%2B+%7B%5Cbf+H%7D%5BY_i%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%5Cfrac%7B1%7D%7Bn%2B1%7D+%5Csum_%7Bi%3D1%7D%5En+%5Csup_%7BY_i%7D+%28n%2B1%29+%7B%5Cbf+E%7D+g_i%28Y_i%29+%2B+%7B%5Cbf+H%7D%5BY_i%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + \frac{1}{n+1} \sum_{i=1}^n \sup_{Y_i} (n+1) {\bf E} g_i(Y_i) + {\bf H}[Y_i]" class="latex" />
where <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> ranges over random variables taking values in <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_0,\dots,X_n}" class="latex" /> range over tuples of random variables taking values in <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%7BY_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BY_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BY_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{Y_i}" class="latex" /> range over random variables taking values in <img src="https://s0.wp.com/latex.php?latex=%7BT_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BT_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BT_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{T_i}" class="latex" />. Comparing the suprema, the claim now reduces to

<blockquote>Lemma 3 (Conditional expectation computation) Let <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> be an <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" />-valued random variable. Then there exists a <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" />-valued random variable <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_n)}" class="latex" />, where each <img src="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_i%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_i}" class="latex" /> has the same distribution as <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%28n%2B1%29+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%28n%2B1%29+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%28n%2B1%29+%7B%5Cbf+H%7D%5BX%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_0,\dots,X_n] = (n+1) {\bf H}[X] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle - {\bf H}[\pi_1(X)] - \dots - {\bf H}[\pi_n(X)]." class="latex" />
</blockquote>


Proof: We induct on <img src="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n}" class="latex" />. When <img src="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn%3D0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n=0}" class="latex" /> we just take <img src="https://s0.wp.com/latex.php?latex=%7BX_0+%3D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_0+%3D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_0+%3D+X%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_0 = X}" class="latex" />. Now suppose that <img src="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn+%5Cgeq+1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n \geq 1}" class="latex" />, and the claim has already been proven for <img src="https://s0.wp.com/latex.php?latex=%7Bn-1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bn-1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bn-1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{n-1}" class="latex" />, thus one has already obtained a tuple <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29+%5Cin+%5COmega_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29+%5Cin+%5COmega_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29+%5Cin+%5COmega_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_{n-1}) \in \Omega_{n-1}}" class="latex" /> with each <img src="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_0,\dots,X_{n-1}}" class="latex" /> having the same distribution as <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />, and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%3D+n+%7B%5Cbf+H%7D%5BX%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_%7Bn-1%7D%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%3D+n+%7B%5Cbf+H%7D%5BX%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_%7Bn-1%7D%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%3D+n+%7B%5Cbf+H%7D%5BX%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_1%28X%29%5D+-+%5Cdots+-+%7B%5Cbf+H%7D%5B%5Cpi_%7Bn-1%7D%28X%29%5D.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_0,\dots,X_{n-1}] = n {\bf H}[X] - {\bf H}[\pi_1(X)] - \dots - {\bf H}[\pi_{n-1}(X)]." class="latex" />
By hypothesis, <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X_{n-1})}" class="latex" /> has the same distribution as <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X)}" class="latex" />. For each value <img src="https://s0.wp.com/latex.php?latex=%7Bt_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bt_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bt_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{t_n}" class="latex" /> attained by <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X)}" class="latex" />, we can take conditionally independent copies of <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_{n-1})}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> conditioned to the events <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X_{n-1}) = t_n}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X%29+%3D+t_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X) = t_n}" class="latex" /> respectively, and then concatenate them to form a tuple <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_n%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_n)}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5COmega_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\Omega_n}" class="latex" />, with <img src="https://s0.wp.com/latex.php?latex=%7BX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_n%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_n}" class="latex" /> a further copy of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> that is conditionally independent of <img src="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_0%2C%5Cdots%2CX_%7Bn-1%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_0,\dots,X_{n-1})}" class="latex" /> relative to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi_n%28X_%7Bn-1%7D%29+%3D+%5Cpi_n%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi_n(X_{n-1}) = \pi_n(X)}" class="latex" />. One can the use the <a href="https://en.wikipedia.org/wiki/Conditional_entropy#Chain_rule">entropy chain rul</a>e to compute <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%7C+%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%7C+%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%5D+%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_n%7C+%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}[X_0,\dots,X_n] = {\bf H}[\pi_n(X_n)] + {\bf H}[X_0,\dots,X_n| \pi_n(X_n)]" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_n%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = {\bf H}[\pi_n(X_n)] + {\bf H}[X_0,\dots,X_{n-1}| \pi_n(X_n)] + {\bf H}[X_n| \pi_n(X_n)] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_%7Bn-1%7D%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_%7Bn-1%7D%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%7C+%5Cpi_n%28X_%7Bn-1%7D%29%5D+%2B+%7B%5Cbf+H%7D%5BX_n%7C+%5Cpi_n%28X_n%29%5D+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = {\bf H}[\pi_n(X)] + {\bf H}[X_0,\dots,X_{n-1}| \pi_n(X_{n-1})] + {\bf H}[X_n| \pi_n(X_n)] " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%28%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_%7Bn-1%7D%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%28%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_%7Bn-1%7D%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D+%7B%5Cbf+H%7D%5B%5Cpi_n%28X%29%5D+%2B+%28%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_%7Bn-1%7D%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle = {\bf H}[\pi_n(X)] + ({\bf H}[X_0,\dots,X_{n-1}] - {\bf H}[\pi_n(X_{n-1})]) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%28%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%28%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%2B+%28%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D%29+&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + ({\bf H}[X_n] - {\bf H}[\pi_n(X_n)]) " class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%2B+%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%2B+%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%3D%7B%5Cbf+H%7D%5BX_0%2C%5Cdots%2CX_%7Bn-1%7D%5D+%2B+%7B%5Cbf+H%7D%5BX_n%5D+-+%7B%5Cbf+H%7D%5B%5Cpi_n%28X_n%29%5D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle ={\bf H}[X_0,\dots,X_{n-1}] + {\bf H}[X_n] - {\bf H}[\pi_n(X_n)]" class="latex" />
and the claim now follows from the induction hypothesis. <img src="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5CBox&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\Box" class="latex" />

With a little more effort, one can replace <img src="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BS%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{S}" class="latex" /> by a more general measure space (and use differential entropy in place of Shannon entropy), to recover Carbery’s inequality in full generality; we leave the details to the interested reader.

]]></content:encoded>
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<media:title type="html">Terry</media:title>
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</item>
<item>
<title>A slightly longer Lean 4 proof tour</title>
<link>https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/</link>
<comments>https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Wed, 06 Dec 2023 07:10:07 +0000</pubDate>
<category><![CDATA[expository]]></category>
<category><![CDATA[math.CA]]></category>
<category><![CDATA[Lean4]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14195</guid>
<description><![CDATA[In my previous post, I walked through the task of formally deducing one lemma from another in Lean 4. The deduction was deliberately chosen to be short and only showcased a small number of Lean tactics. Here I would like to walk through the process I used for a slightly longer proof I worked out […]]]></description>
<content:encoded><![CDATA[
In my <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous post</a>, I walked through the task of formally deducing one lemma from another in <a href="https://en.wikipedia.org/wiki/Lean_(proof_assistant)">Lean 4</a>. The deduction was deliberately chosen to be short and only showcased a small number of Lean tactics. Here I would like to walk through the process I used for a slightly longer proof I worked out recently, after seeing the following <a href="https://twitter.com/damekdavis/status/1730983634570510819">challenge from Damek Davis</a>: to formalize (in a civilized fashion) the proof of the following lemma:
<blockquote class="wp-block-quote is-layout-flow wp-block-quote-is-layout-flow">
Lemma. Let <img src="https://s0.wp.com/latex.php?latex=%5C%7Ba_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C%7Ba_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C%7Ba_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\{a_k\}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%5C%7BD_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C%7BD_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C%7BD_k%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\{D_k\}" class="latex" /> be sequences of real numbers indexed by natural numbers <img src="https://s0.wp.com/latex.php?latex=k%3D0%2C1%2C%5Cdots&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=k%3D0%2C1%2C%5Cdots&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=k%3D0%2C1%2C%5Cdots&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="k=0,1,\dots" class="latex" />, with <img src="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k" class="latex" /> non-increasing and <img src="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_k" class="latex" /> non-negative. Suppose also that <img src="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+D_k+-+D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+D_k+-+D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k+%5Cleq+D_k+-+D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k \leq D_k - D_{k+1}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=k+%5Cgeq+0&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=k+%5Cgeq+0&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=k+%5Cgeq+0&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="k \geq 0" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+%5Cfrac%7BD_0%7D%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k+%5Cleq+%5Cfrac%7BD_0%7D%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k+%5Cleq+%5Cfrac%7BD_0%7D%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k \leq \frac{D_0}{k+1}" class="latex" /> for all <img src="https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="k" class="latex" />.
</blockquote>
Here I tried to draw upon the lessons I had learned from the PFR formalization project, and to first set up a human readable proof of the lemma before starting the Lean formalization – a lower-case “blueprint” rather than the fancier <a href="https://github.com/PatrickMassot/leanblueprint">Blueprint</a> used in the PFR project. The main idea of the proof here is to use the telescoping series identity
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D+%3D+D_0+-+D_%7Bk%2B1%7D.&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D+%3D+D_0+-+D_%7Bk%2B1%7D.&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D+%3D+D_0+-+D_%7Bk%2B1%7D.&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i=0}^k D_i - D_{i+1} = D_0 - D_{k+1}." class="latex" />
Since <img src="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_{k+1}" class="latex" /> is non-negative, and <img src="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_i \leq D_i - D_{i+1}" class="latex" /> by hypothesis, we have
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+a_i+%5Cleq+D_0&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+a_i+%5Cleq+D_0&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Csum_%7Bi%3D0%7D%5Ek+a_i+%5Cleq+D_0&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \sum_{i=0}^k a_i \leq D_0" class="latex" />
but by the monotone hypothesis on <img src="https://s0.wp.com/latex.php?latex=a_i&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_i&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_i&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_i" class="latex" /> the left-hand side is at least <img src="https://s0.wp.com/latex.php?latex=%28k%2B1%29+a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%28k%2B1%29+a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%28k%2B1%29+a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="(k+1) a_k" class="latex" />, giving the claim.
This is already a human-readable proof, but in order to formalize it more easily in Lean, I decided to rewrite it as a chain of inequalities, starting at <img src="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k" class="latex" /> and ending at <img src="https://s0.wp.com/latex.php?latex=D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_0 / (k+1)" class="latex" />. With a little bit of pen and paper effort, I obtained
<img src="https://s0.wp.com/latex.php?latex=a_k+%3D+%28k%2B1%29+a_k+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k+%3D+%28k%2B1%29+a_k+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k+%3D+%28k%2B1%29+a_k+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k = (k+1) a_k / (k+1)" class="latex" />
(by field identities)
<img src="https://s0.wp.com/latex.php?latex=%3D+%28%5Csum_%7Bi%3D0%7D%5Ek+a_k%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3D+%28%5Csum_%7Bi%3D0%7D%5Ek+a_k%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3D+%28%5Csum_%7Bi%3D0%7D%5Ek+a_k%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="= (\sum_{i=0}^k a_k) / (k+1)" class="latex" />
(by the formula for summing a constant)
<img src="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+a_i%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+a_i%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+a_i%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq (\sum_{i=0}^k a_i) / (k+1)" class="latex" />
(by the monotone hypothesis)
<img src="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq+%28%5Csum_%7Bi%3D0%7D%5Ek+D_i+-+D_%7Bi%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq (\sum_{i=0}^k D_i - D_{i+1}) / (k+1)" class="latex" />
(by the hypothesis <img src="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_i+%5Cleq+D_i+-+D_%7Bi%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_i \leq D_i - D_{i+1}" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%3D+%28D_0+-+D_%7Bk%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3D+%28D_0+-+D_%7Bk%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3D+%28D_0+-+D_%7Bk%2B1%7D%29+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="= (D_0 - D_{k+1}) / (k+1)" class="latex" />
(by telescoping series)
<img src="https://s0.wp.com/latex.php?latex=%5Cleq+D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq+D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq+D_0+%2F+%28k%2B1%29&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq D_0 / (k+1)" class="latex" />
(by the non-negativity of <img src="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_%7Bk%2B1%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_{k+1}" class="latex" />).
I decided that this was a good enough blueprint for me to work with. The next step is to formalize the statement of the lemma in Lean. For this quick project, it was convenient to use the <a href="https://live.lean-lang.org/">online Lean playground</a>, rather than my local IDE, so the screenshots will look a little different from those in the previous post. (If you like, you can follow this tour in that playground, by clicking on the screenshots of the Lean code.) I start by importing Lean’s math library, and starting an example of a statement to state and prove:
<img src="image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATkAAACaCAYAAADSHj8FAAAYwElEQVR4Ae2dX2wcx33H992vQbGvtPUiWyjEAAV4D0WhQ1hALBQgRPlgNnog+ESwbuDSQu0QQmOGISCZVQGFNWKBUirFx9gSSDtCKLRCT1Fh9GhAxlGJLcqp3CMKtjhXMkRTgnyKSflbzO3M7uzc7t7ece94t/oKONz+nZ35zO6Hv/mzJ+urr74CP2TAe4D3QFrvAevx48fghwx4D/AeSOs9YH399dfghwx4D/AeSOs9YG1vb4MfMuA9wHsgrfeAtbOzA37IgPcA74G03gPWkydPwA8Z8B7gPZDWe8D65ptvwA8Z8B7gPZDWe4CSo+T5R473QKrvAUqON3iqb/C0RicsV/zI2wL/kQAJkECKCVByKa5cFo0ESACg5HgXkAAJpJoAJZfq6mXhSIAE0i25MpA7DcxfY0WTQDoJbN5ZxvzJOSx+vJnOAiZQqvRKrgJM2oBlOZ/xKwnQYhIkgDJyQxYsy8LgQrkJHt75Ig3fZ6rQWHp35jHgppHB3MfRp5cXBp3rDeXQTM6jU+/cvXsjuS9+h/k3z+D5V3+K5149j4t3WwCoDAxLwQnRDS604BppSvIeMH8cGOwDGnzUmqJQmPIe8NGl4Chkc2nUk0CjAoiVqwpKl+cwMZLF6MW4j70nqWYlt/xyFtl+77PfliwaLeONadiu5CxMr0QXmpKL5pPM3gef4eLPz6K3KjchuBZKDkDxNNBjAT1DQPFhMkVIbSorXtTbbslZI4uo1Zwnk2q0c3SxBdGHd434wmrmnKi7xkvPalRyKEEI07ZsZF9eRinqMgAouTqAdrX7wQZ+c+m8K7fn37iMi++ebbnkdpXnp+3ktkqugvwxEb30oveg+B5EbsMALptitm23sInlCaY7JWcwq7NKydUBtJvdWx+87Qjth2fwav4zbD0B7ubPU3K7gZr0uW2VnJLLIKannH6igXP+OKR4qrcqwdmTk5RcQnVNySUEMjCZP9zCP715GYUvdtzdLZHcBjCo9cOpQQfxHdlfIR/wwQtO9paPOc1c+wCwuO5sK10GBvYBlg0MnwYqbkmchWl5XRWRrC0Bo4cAW27PjADLMi3jVG91Gygs+M8TeRg9BuQjuoxyQ05T0+13LANzY0BG5Fc01/uAiQX4m4QRrHRu+nIkQ68UMZY8yeWuyc7zw/Nac6uIWRHhHZxFcXUWvaLfqV/fLy9R2cTalXlMjg0i29fj9t/ZB7IYHJvD8p3aRrDeF+jr9Nf6tnzbazrpVd69gYfytTmMH9kv+8d6kBmZxuJt8w4Jw+KlF6u5ujLtltOXz6Bo2LhkjeTKBeSmRpHZ5/QJ2gcGMH6u6L9PjDS6cXVvBh7Qokgu4sGNfEBVFDMFFKa8vqnqAz4GlFaAjCHP2VV/dbuS+xiYPWKkoc61gdkb/vPctXVgVEpJF4u+PBoyeKJLbu2cI2j9PLWcmdLkHMFKHW9+RzJ0CxJnQT3Yoplawvxh8ZBlMX9HnrvidKj3niwCGzkMVgU0XTMgUl9YPRi96I8Q65/jPPCuQCIlt4bCjOgTM86prmcwfSOO6BQLC+2U3NrKNLJqwMPIf89IK/o/49wXrTkmXZIzGWkPcuQDqiTX58hsQkw3eQiMCznZwOBhwD6KaqSx/JIjsOwZ/8WU5Gw5bSXzElC85xxT2QAm+6T4Djvp+M6+B4zL8+wjwPJtb29lHZiVkZqQzuQH3j61pCTX2+dEjvYhILcqhSaiw5OedF2RqJPVt2JgtWN0tYjp6gPm9MWVzg1Uo5PeU0WIXOdfE/1wUnoRkiueGsD46UUUbpexqfmksr6MyT4pHnscy7UBnSy1J5hm+uRUf2HPyDyK92QGytq1awSpYOvfXh5iSU4/VSy7kV1Av6ZxrBvJHcwgI/jbA5i+VkJlG8D2JopnhqWwbUxc1YAa6XTbKiVXvVE8CdhjXrNuvt/bviiFVV6Q26b8Va0kJ0Q0cEqLmNRhq0CvjOjMKNCNHsX0jZBRYCUyK0CS7j7RNB0BSuKm1f9VgAl57eGL+g5tua2SK2C6Gj3IB/PeIkbFumieVvKYEMuq+RohOS33tYvaHLLwB9YTTDOSE9Fe5qXaUc3K1QnZpIwSrMqyl4e2SU7w7RsP6D7ZxOKI/OMQOOKt8txd35ScqC/tAdcjHSWPzCmvUutJztck9E6rLs0edATpiwI1AY0uGSdoq5VrXv+enkdxiMqnNRQQJco01DFuv52WdnVRY9D6KSRKcqoJqh6uXkweH68KwpWOK7lhLEb0S5rFEfGoI1Kv76z2GE8w7vVqDzK2eOfYR3NaP6J2mJvn+tEVtMnFbZOcPYxcSP/w5mWHv2VNIJ+SYI6SE/dmyAOuxKA3detJTj9Wu+2ri8tjAVGgFuGpQQvzvOr6pmw+BzRZVT4tI7oMTCdsYwiDsMN3tV1Fa5aSHOBFPyKSGIWKnFFexLAe9cW+cOslFyrGTpdcVDNaDfTEGMiIXRV7fCAlJyog5AFX8tDFtRvJuc1SrUkcdu2g+0I1ic1oTOWzayTnSsCTHCp5TKqO8LFlbYRPySokKrq3hvzCHGaPjfreIsj2Z9BTlWPrIrlUSs6tm16Y3SpB92Q3bKPk9kJyx7VbI0Sw2hHuYqolBzGyLQYcbExe09tJSnLa6GuVSAVr50ZdkbmjoVJs+nqojLSmYvgxLn654DVXQ89xRREiZl+SXnpta65GRXJ63kOatL7sd8EKJScqKUQ0KkJKKpJbPOo0V+0Z7c7Qrh23uarnR6Sk8tnVkZyGxL+oJOd/N1M0b52pGzYGZvIo6cOr1QS880JlRMn5UYs1933YOIMmtad34hZKTtSKJhq9013JQ5dK081VbYBh4qp2K2h9bb7t2iFiUR94MKdEqHwmJbnGOviNjMZZdac9aM3V0PM8WXn1sInlsXqjgN55cSRnz4jpK3H+eZFXaLp6NGS+rlZzCS+9TojkSmeyzsiwGt2uyW/3baDkRJ21QXKlC3LQ4SBQ0FtjABZH5L6A6SHqlpo/LKPA12qnpyQiuXXvbZHJVv/+npJcVLNJFVwbJfUkp4nhuP5nyT0J5SuTyNTtkwMKx6UsYz/U3rVTJ7mHBUxW3yWO6sf0GHfLUnsk9+Qxth5s+T63fq1e0D+LX/yXvu8RHj9JCF+jk4GNibBKHt7DJX7JQQrJGMlU/WXirQTDYSgteW9MjF8OKNuqt19MBs5rfSFiMvC0eoPCBhYDIgOVz11FcgCUSK0+bTJxQHZ3vakhyXlSGdZ+DsnpvxMTWocxv6pm+1aweXsZs0PeK16iby5URtUIedLf7A2Zp+iV2ctPaLqdHsn1z6Ig53265bpXxPxR+WMIsYXvnt3RC+2R3N3rGPX9vJL6maWg7+Z/X84dvVSvUUV86+JKMpJTr0Jl+oFsP7Bf++HOsNeyxB0i3o01Xx1TaVW/9wHz2psQ+l2VlOQqAa+v6XnwMdMz0OCyOxcrViQXIpX1HIbVaGzAYEPP0BxmX3KitFAZVfNdQk493AHpWDV5DMmPziBKckrwQdcyt9X89JJ3bX1gJXi5dtDDfeNBXWdfxhmR1t/7PTSNgvqboZepi5cpOVF5CTZXew54k3aFIMTL8eMngUKciazi59qngOwBGS1awP5DwMQ5oGyGh9pNl5TkqknKl/v1PCjRJSU592GrEYhWKHfRe7BrZFXOY25s0H3B3LL3IzsyjdyKA1u9p1pznpu2WtjE2tI0RrVpJ644avIYkR+VXIdKDtsVlFZymBPTbQ6pHxQQ0fB+ZIfGMXelpE3dUYXp/u/2SK77OdUtgWquJiWCuhfkASRAArEIUHKxMNU/iJKrz4hHkMBeEKDkEqJOySUEksmQQMIEKLmEgFJyCYFkMiSQMAFKLiGglFxCIJkMCSRMgJJLGCiTIwES6CwClFxn1QdzQwIkkDABSi5hoEyOBEigswhQcp1VH8wNCZBAwgQouYSBMjkSIIHOIkDJdVZ9MDckQAIJE6DkEgbK5EiABDqLACXXWfXB3JAACSRMgJJLGCiTIwES6CwClFxn1QdzQwIkkDABSi5hoEyOBEigswhQcp1VH8wNCZBAwgQouYSBMjkSIIHOIkDJdVZ9MDckQAIJE6DkEgbK5EiABDqLQHsl92ADv3n/HXz/x97/0tX3k7cxk/897ib13xB2Fl/mhgRIYI8JtElyO9j44B0c+aEnt+eM/6Lw+Z9cRjHif6TaY068PAmQQJcSaI/k/vtfcURK7fuXbuCzL3ccXNtbuJV/B38u9/1Z7hYedylIZpsESKAzCbRHcgA28u9g5qP7ARQeo5h7E05k9zb+5UHAIdxEAiRAAk0SaJvkIvP3yXtScj/FP6xFHsmdJEACJNAQgY6T3MwnDeWfB5MACZBAJIHOkNxNFcmdxcXPI/PLnSRAAiTQEIEOkNxjFM7LUdcTeWw0lH0eTAIkQALRBPZcco8/ec8dXX35P4IGJqILwL0kQAIkEEVgbyX3+XWMyrlzz/9sBVtROeU+EiABEmiCwN5J7sHvMCPffOBE4CZqjqeQAAnEIrA3ktME99yP3+ObDrGqigeRAAk0Q6D9krtfxN8zgmumrngOCZBAEwTaKzkKrokq4ikkQAK7IdA+yemCe+MKbvFl/N3UG88lARKISaA9ktME99yJyyh8voWtB+Gfx9sxc8/DSIAESKAOgbZI7m7+vPtuqvkTS0HrfLWrTq1xNwmQQGwClFxsVDyQBEigGwm0RXLdCIZ5JgESSAcBSi4d9chSkAAJhBCg5ELAcDMJkEA6CFBy6ahHloIESCCEACUXAoabSYAE0kGAkktHPbIUJEACIQQouRAw3EwCJJAOApRcOuqRpSABEgghQMmFgOFmEiCBdBCg5NJRjywFCZBACIFUSO7+/fvgJ/0MLMuC/mGdp7/Ok6hjSo6C7Jo/ELrgxHISDwDTSL8oKTlKrmtkQcmlX0it+KNDyVFylBzvga65B5qRYColF9L/yM1dTsCM5Lq8OMx+mwhQcm0CzcvsngAlt3uGT2MKlNzTWOtdWmZKrksrbo+zTcntcQXw8vEJUHLxWfFIjwAl57HgUocToOQ6vII6NHuUXIdWTNuytQEMWoAV8JleaVsuYl2IkouFiQcZBNonua0NXL90AS8fPY5vv/ADPNPzAzzzwisYOPqPeOvfNvDljpGzBlbNYeUGTuWhZWCiH8iqzyHAlsKj5Hh7pIFAmyT3Od4dkWITcgv4PDv0S9x81BxSSq45boFnaZEdJRdIiBu7jEDbJPerv5nBiUsf4dPyVx6iRxv48K038IKUXvbN//T2NbBEyTUAq96hlFw9QtzfZQTaJLkoKl/j+o9U8/UCbkYdGrKPkgsB08xmSq4Zajyngwl0gOSAzy/NyCbsWXzYBKwayW0D+dPAYJ/Xob7/EDC9BGyGpD+rjrWB5XvBB21e9vqrhhdqj6msA7nTwGg/sN/2rp3pB6YXgPJ27TnlBee4yQ8APATmXnSu0XMYKDx0ji+eluntAyYuG2msyOsMAWWxaxNYnAKyB+T2fcDoFFAKuLaRkrPapOTK14DxIaBH9ufZB5zrroUBD7x49EYOPETz4d5gAp0luRcSiORubmN0n3zAbdmhrgRmAfYhTx4+JOvAsBSTPRYgw3vAuNp/DKj4TgagySFopFJsC7q2ktzgBSB3VOZbiqL3FFBa8MRaTdeUsJLc94C1VWBAk6ueD3HtYk2mzUL4yxGrT24byI14+RZ/TMQghpKdZQOx0gnIirmJkjOJcD0OgQ6Q3BZ+Ne40V5/90Ue18ohRCjeS2/ga43/kPHCZ14zIqQxMStkFSgx+oQxf9F94eUw+yH0hktwAxkeA3BWgpEcvFaB4xhNV5rQ/XSW53j7nmLnbAG4DWSG6w8CgDWSmHKnO9Tt5mLiqpaEkJyQqBTe+AGxKoZWvABkpzYEz2nlhi5qs48gpf8zjslwNJWXC28Dya3KfKeawa9fZTsnVAcTdgQT2VnI7X+D6qSk8KwYevnMWH+5qdPVLXH9dPlSHgVJQce8AA/KBn10NOgBwH1obWNxwjtGbqdM3gs+rt9VNt9+fNyU5EXVlTslUKsCEzKd1ECjKzYUpp3yDelNZk5yImoLKVTwpuWhphea3EcmtAr0yn/N3glOcP+xcu/dk8P5GtlJyjdDisYpAmyV3C2999zgGqp9XvVHVY1fxaZOCEwWpRnLlP2BCRnG+SEeVVH7nvuc8dNmwqOYhMK0ivqNASWvGiiZls/90mRW0RNztNpCXfXBi97SUx7jWB1dPcqGRVwwZuVlqQHJ5FakFNd9lgqULUrCG3N3rNbBAyTUAi4e6BNosud/iRMAcuWd6XsHg376Pm1tuvhpaqEru33fwx1UxfIOcjMCCElGisCIezMoNr4nXI/v3bCG8oARjbnNlZgGBklMDBzI9JTm9LCrvYZGcnq4vW5vAuJRmqAjVCQ1IbvagIzBfflQ66luLNPNx+gTVeQHflFwAFG6qS6DNkjPy8+gLfPrB+3jtL9QUkjfwboSgjLPd1arkru64ryapzm93Fr+azd8PZNSghCEVNzG54EYgQg5GlGUeq6+XV4H5k85Io+/6arRzLyQXEhnq+XaXG5CcErEYSfWVVeOd1QZ9dGG712tggZJrABYPdQnsreRUNh79Fie+0/zggyk5fVQxdDloBFXlB4CKmqrnhw02aMfjHjB5SDbNZNQUdm094nIjPEO6SiC6GFSefJGTFinp6epZE8sqvepUFXOnvt6E5MLKaW5f1gdk9GvGXKbkYoLiYT4CnSE5AOv/fFy+z9r4NBK/5KKbq77Sh6xUVrzmqhqxrNdcVR3sVh+QWwUqxrw0V2Z7EcmVgWHVXK03cNKE5HzSDWGaxGZKLgmKT18aHSO5m2++6khu+NdYb7AeqpL7dBt/qR7k3fx6hjbQIKZubGr9c+7op5k/rWM/aHRTHL6Xkqtc9SLMuv1iDUhuUc7rs6dMIK1Zp+RawzXtqXaG5B59hNf/1GmuDr7deKdcVXL3H+H8XzkPc72oK6pSc0NSCFoTtXTGk0Rgx73WZMwFGXoTmNcm+urNSld+LWyuqjL1yvl2UeXXJzUHllU7eXNJcrGBwHJrxyaxSMklQfHpS2NPJVfZ2sDNy7/Ei9+WAw9NzpVzJHcf5fwO/kRGc/YRYPm2v9lY2QSKS8DEi97cM73KVZ+X6Esy58O5byMEPdBa9FOdhKxGER8ChXNA1ngLoVWS0/vvquWqAIsveSIKe11NZ9CI5MTMbTXdpvpmwxWgrE2DEa+plcRAzDFgtl4z2ZeJ4BVKLpgLt0YTaI/kbpwN/Hkl/SeXXjh6aXdTSKr/rdyXWF144s7wNzu+3XW7VnKiSad+Ry2wWao1Y8WbCOZ0El2Q7nXUAIQNTJzyfpyyVZKrXnefHOnUfhfO2hceaUXmW+U/qhtgHRjXRlBryq7OpeSin0TubRmBPZTcK8h+dwYvn7qK6581OUFOYlGRnPoWb6rnTgKDxoOeHQHml4CyirQUVl1gQ7UCU4eVL2oiDGj6rS0Bo9o1e/qA8dPyNS8t2muV5PZr01SEbDJDwGzIDwOoMu1aciKhbaCw4EybMX+YYOIcUGy8B0Jlz/fNSM6HgysxCbRHcjEz0+xhSm7qu9l0uvI8rT9Ql2dXlqVOpim5OoC4O5AAJReIpYs2UnJdVFnM6l4QoOT2gnqS16TkkqTJtFJIgJLr9kql5Lq9Bpn/FhOg5FoMuOXJP8WS+99vWXh08RctR8wLdDcBSq676y+VuQ8TlznwICRH0aXyFki0UKmUnBpl5fd957f2qnMIu2dZiOvuz3/my7tYNyX3f6//XVVyQcez7runvltdV5Rclwmg1TdEJ6SvIjQlOvEttpmSE3ml6CizevcsJUfJ+SKmejdMO/br4ir/9YgbrQVJzhSdOL4deeQ1ukeulBwl15FS0EX3P9+yqhFbmORM0VFA3SOgdtRVKiSXaC8lEyMBEkgVAUouVdXJwpAACZgEKDmTCNdJgARSRYCSS1V1sjAkQAImAUrOJMJ1EiCBVBGg5FJVnSwMCZCASYCSM4lwnQRIIFUEKLlUVScLQwIkYBKg5EwiXCcBEkgVAUouVdXJwpAACZgEKDmTCNdJgARSRYCSS1V1sjAkQAImAUrOJMJ1EiCBVBGg5FJVnSwMCZCASYCSM4lwnQRIIFUEKLlUVScLQwIkYBKg5EwiXCcBEkgVAUouVdXJwpAACZgEKDmTCNdJgARSRYCSS1V1sjAkQAImAUrOJMJ1EiCBVBGg5FJVnSwMCZCASYCSM4lwnQRIIFUEKLlUVScLQwIkYBKg5EwiXCcBEkgVAUouVdXJwpAACZgEKDmTCNdJgARSRYCSS1V1sjAkQAImAUrOJMJ1EiCBVBGg5FJVnSwMCZCASYCSM4lwnQRIIFUEKLlUVScLQwIkYBKg5EwiXCcBEkgVAUouVdXJwpAACZgEKDmTCNdJgARSReD/ARa8RcYpet5CAAAAAElFTkSuQmCC" alt=""/>
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)"><img width="256" height="124" data-attachment-id="14279" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-26-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-26.png" data-orig-size="256,124" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-26" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-26.png?w=256" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-26.png?w=256" src="https://terrytao.files.wordpress.com/2023/12/image-26.png?w=256" alt="" class="wp-image-14279" srcset="https://terrytao.files.wordpress.com/2023/12/image-26.png 256w, https://terrytao.files.wordpress.com/2023/12/image-26.png?w=150 150w" sizes="(max-width: 256px) 100vw, 256px" /></a></figure>
Now we have to declare the hypotheses and variables. The main variables here are the sequences <img src="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="a_k" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_k" class="latex" />, which in Lean are best modeled by functions <code>a</code>, <code>D</code> from the natural numbers ℕ to the reals ℝ. (One can choose to “hardwire” the non-negativity hypothesis into the <img src="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=D_k&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="D_k" class="latex" /> by making <code>D</code> take values in the nonnegative reals <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbf+R%7D%5E%2B&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbf+R%7D%5E%2B&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbf+R%7D%5E%2B&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="{\bf R}^+" class="latex" /> (denoted <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Real/NNReal.html#NNReal">NNReal</a></code> in Lean), but this turns out to be inconvenient, because the laws of algebra and summation that we will need are clunkier on the non-negative reals (which are not even a group) than on the reals (which are a field). So we add in the variables:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)"><img width="452" height="159" data-attachment-id="14205" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-26/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image.png" data-orig-size="452,159" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image.png?w=452" src="https://terrytao.files.wordpress.com/2023/12/image.png?w=452" alt="" class="wp-image-14205" srcset="https://terrytao.files.wordpress.com/2023/12/image.png 452w, https://terrytao.files.wordpress.com/2023/12/image.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image.png?w=300 300w" sizes="(max-width: 452px) 100vw, 452px" /></a></figure>
Now we add in the hypotheses, which in Lean convention are usually given names starting with <code>h</code>. This is fairly straightforward; the one thing is that the property of being monotone decreasing already has a name in Lean’s Mathlib, namely <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Basic.html#Antitone">Antitone</a></code>, and it is generally a good idea to use the Mathlib provided terminology (because that library contains a lot of useful lemmas about such terms).
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)"><img width="1024" height="121" data-attachment-id="14289" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-27/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-27.png" data-orig-size="1519,180" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-27" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=1024" alt="" class="wp-image-14289" srcset="https://terrytao.files.wordpress.com/2023/12/image-27.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-27.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-27.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-27.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-27.png 1519w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
One thing to note here is that Lean is quite good at filling in implied ranges of variables. Because <code>a</code> and <code>D</code> have the natural numbers ℕ as their domain, the dummy variable <code>k</code> in these hypotheses is automatically being quantified over ℕ. We could have made this quantification explicit if we so chose, for instance using <code>∀ k : ℕ, 0 ≤ D k</code> instead of <code>∀ k, 0 ≤ D k</code>, but it is not necessary to do so. Also note that Lean does not require parentheses when applying functions: we write <code>D k</code> here rather than <code>D(k)</code> (which in fact does not compile in Lean unless one puts a space between the <code>D</code> and the parentheses). This is slightly different from standard mathematical notation, but is not too difficult to get used to.
This looks like the end of the hypotheses, so we could now add a colon to move to the conclusion, and then add that conclusion:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)"><img loading="lazy" width="1024" height="128" data-attachment-id="14209" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-1-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-1.png" data-orig-size="1491,187" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-1" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=1024" alt="" class="wp-image-14209" srcset="https://terrytao.files.wordpress.com/2023/12/image-1.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-1.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-1.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-1.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-1.png 1491w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
This is a perfectly fine Lean statement. But it turns out that when proving a universally quantified statement such as <code>∀ k, a k ≤ D 0 / (k + 1)</code>, the first step is almost always to open up the quantifier to introduce the variable <code>k</code> (using the Lean command <code>intro k</code>). Because of this, it is slightly more efficient to hide the universal quantifier by placing the variable <code>k</code> in the hypotheses, rather than in the quantifier (in which case we have to now specify that it is a natural number, as Lean can no longer deduce this from context):
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)"><img loading="lazy" width="1024" height="117" data-attachment-id="14212" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-2-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-2.png" data-orig-size="1628,187" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-2" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=1024" alt="" class="wp-image-14212" srcset="https://terrytao.files.wordpress.com/2023/12/image-2.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=1019 1019w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-2.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-2.png 1628w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
At this point Lean is complaining of an unexpected end of input: the example has been stated, but not proved. We will temporarily mollify Lean by adding a <code>sorry</code> as the purported proof:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20sorry"><img loading="lazy" width="1024" height="114" data-attachment-id="14214" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-3-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-3.png" data-orig-size="1622,182" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-3" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=1024" alt="" class="wp-image-14214" srcset="https://terrytao.files.wordpress.com/2023/12/image-3.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=1016 1016w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-3.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-3.png 1622w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Now Lean is content, other than giving a warning (as indicated by the yellow squiggle under the <code>example</code>) that the proof contains a sorry.
It is now time to follow the blueprint. The Lean tactic for proving an inequality via chains of other inequalities is known as <code><a href="https://leanprover-community.github.io/extras/calc.html">calc</a></code>. We use the blueprint to fill in the <code>calc</code> that we want, leaving the justifications of each step as “<code>sorry</code>”s for now:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%20%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="302" data-attachment-id="14222" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-6-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-6.png" data-orig-size="1628,481" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-6" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=1024" alt="" class="wp-image-14222" srcset="https://terrytao.files.wordpress.com/2023/12/image-6.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/12/image-6.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-6.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-6.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-6.png 1628w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Here, we “<code>open</code>“ed the <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Basic.html">Finset</a></code> namespace in order to easily access <code>Finset</code>‘s <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Finset/Basic.html#Finset.range">range</a></code> function, with <code>range k</code> basically being the finite set of natural numbers <img src="https://s0.wp.com/latex.php?latex=%5C%7B0%2C%5Cdots%2Ck-1%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C%7B0%2C%5Cdots%2Ck-1%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C%7B0%2C%5Cdots%2Ck-1%5C%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\{0,\dots,k-1\}" class="latex" />, and also “<code>open</code>“ed the <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html">BigOperators</a></code> namespace to access the familiar ∑ notation for (finite) summation, in order to make the steps in the Lean code resemble the blueprint as much as possible. One could avoid opening these namespaces, but then expressions such as <code>∑ i in range (k+1), a i</code> would instead have to be written as something like <code>Finset.sum (Finset.range (k+1)) (fun i ↦ a i)</code>, which looks a lot less like like standard mathematical writing. The proof structure here may remind some readers of the “two column proofs” that are somewhat popular in American high school geometry classes.
Now we have six sorries to fill. Navigating to the first sorry, Lean tells us the ambient hypotheses, and the goal that we need to prove to fill that sorry:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-5.png"><img loading="lazy" width="672" height="235" data-attachment-id="14219" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-5-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-5.png" data-orig-size="672,235" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-5" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-5.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-5.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-5.png?w=672" alt="" class="wp-image-14219" srcset="https://terrytao.files.wordpress.com/2023/12/image-5.png 672w, https://terrytao.files.wordpress.com/2023/12/image-5.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-5.png?w=300 300w" sizes="(max-width: 672px) 100vw, 672px" /></a></figure>
The ⊢ symbol here is Lean’s marker for the goal. The uparrows ↑ are coercion symbols, indicating that the natural number <code>k</code> has to be converted to a real number in order to interact via arithmetic operations with other real numbers such as <code>a k</code>, but we can ignore these coercions for this tour (for this proof, it turns out Lean will basically manage them automatically without need for any explicit intervention by a human). 
The goal here is a self-evident algebraic identity; it involves division, so one has to check that the denominator is non-zero, but this is self-evident. In Lean, a convenient way to establish algebraic identities is to use the tactic <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#field_simp">field_simp</a></code> to clear denominators, and then <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#ring">ring</a></code> to verify any identity that is valid for commutative rings. This works, and clears the first <code>sorry</code>:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%20%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="299" data-attachment-id="14223" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-7-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-7.png" data-orig-size="1630,476" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-7" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=1024" alt="" class="wp-image-14223" srcset="https://terrytao.files.wordpress.com/2023/12/image-7.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-7.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-7.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-7.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-7.png 1630w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
<code>field_simp</code>, by the way, is smart enough to deduce on its own that the denominator <code>k+1</code> here is manifestly non-zero (and in fact positive); no human intervention is required to point this out. Similarly for other “clearing denominator” steps that we will encounter in the other parts of the proof.
Now we navigate to the next `sorry`. Lean tells us the hypotheses and goals:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-8.png"><img loading="lazy" width="772" height="277" data-attachment-id="14226" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-8-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-8.png" data-orig-size="772,277" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-8" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-8.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-8.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-8.png?w=772" alt="" class="wp-image-14226" srcset="https://terrytao.files.wordpress.com/2023/12/image-8.png 772w, https://terrytao.files.wordpress.com/2023/12/image-8.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-8.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-8.png?w=768 768w" sizes="(max-width: 772px) 100vw, 772px" /></a></figure>
We can reduce the goal by canceling out the common denominator <code>↑k+1</code>. Here we can use the handy Lean tactic <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#congr">congr</a></code>, which tries to match two sides of an equality goal as much as possible, and leave any remaining discrepancies between the two sides as further goals to be proven. Applying <code>congr</code>, the goal reduces to
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-9.png"><img loading="lazy" width="768" height="226" data-attachment-id="14229" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-9-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-9.png" data-orig-size="768,226" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-9" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-9.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-9.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-9.png?w=768" alt="" class="wp-image-14229" srcset="https://terrytao.files.wordpress.com/2023/12/image-9.png 768w, https://terrytao.files.wordpress.com/2023/12/image-9.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-9.png?w=300 300w" sizes="(max-width: 768px) 100vw, 768px" /></a></figure>
Here one might imagine that this is something that one can prove by induction. But this particular sort of identity – summing a constant over a finite set – is already covered by Mathlib. Indeed, searching for <code>Finset</code>, <code>sum</code>, and <code>const</code> soon leads us to the <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html#Finset.sum_const">Finset.sum_const</a></code> lemma here. But there is an even more convenient path to take here, which is to apply the powerful tactic <code>simp</code>, which tries to simplify the goal as much as possible using all the “<code>simp</code> lemmas” Mathlib has to offer (of which <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html#Finset.sum_const">Finset.sum_const</a></code> is an example, but there are thousands of others). As it turns out, <code>simp</code> completely kills off this identity, without any further human intervention:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%20%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="318" data-attachment-id="14233" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-10-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-10.png" data-orig-size="1645,511" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-10" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=1024" alt="" class="wp-image-14233" srcset="https://terrytao.files.wordpress.com/2023/12/image-10.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-10.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-10.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-10.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-10.png 1645w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Now we move on to the next sorry, and look at our goal:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-11.png"><img loading="lazy" width="785" height="267" data-attachment-id="14235" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-11-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-11.png" data-orig-size="785,267" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-11" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-11.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-11.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-11.png?w=785" alt="" class="wp-image-14235" srcset="https://terrytao.files.wordpress.com/2023/12/image-11.png 785w, https://terrytao.files.wordpress.com/2023/12/image-11.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-11.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-11.png?w=768 768w" sizes="(max-width: 785px) 100vw, 785px" /></a></figure>
<code>congr</code> doesn’t work here because we have an inequality instead of an equality, but there is a powerful relative <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Tactic/GCongr/Core.html">gcongr</a></code> of <code>congr</code> that is perfectly suited for inequalities. It can also open up sums, products, and integrals, reducing global inequalities between such quantities into pointwise inequalities. If we invoke <code>gcongr with i hi</code> (where we tell <code>gcongr</code> to use <code>i</code> for the variable opened up, and <code>hi</code> for the constraint this variable will satisfy), we arrive at a greatly simplified goal (and a new ambient variable and hypothesis):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-12.png"><img loading="lazy" width="762" height="244" data-attachment-id="14239" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-12-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-12.png" data-orig-size="762,244" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-12" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-12.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-12.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-12.png?w=762" alt="" class="wp-image-14239" srcset="https://terrytao.files.wordpress.com/2023/12/image-12.png 762w, https://terrytao.files.wordpress.com/2023/12/image-12.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-12.png?w=300 300w" sizes="(max-width: 762px) 100vw, 762px" /></a></figure>
Now we need to use the monotonicity hypothesis on <code>a</code>, which we have named <code>ha</code> here. Looking at the <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Basic.html">documentation for Antitone</a>, one finds a lemma that looks applicable here:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-13.png"><img loading="lazy" width="1024" height="166" data-attachment-id="14241" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-13-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-13.png" data-orig-size="1560,253" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-13" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=1024" alt="" class="wp-image-14241" srcset="https://terrytao.files.wordpress.com/2023/12/image-13.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-13.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-13.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-13.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-13.png 1560w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
One can apply this lemma in this case by writing <code>apply Antitone.imp ha</code>, but because <code>ha</code> is already of type <code>Antitone</code>, we can abbreviate this to <code>apply ha.imp</code>. (Actually, as indicated in the documentation, due to the way <code>Antitone</code> is defined, we can even just use <code>apply ha</code> here.) This reduces the goal nicely:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-14.png"><img loading="lazy" width="675" height="255" data-attachment-id="14244" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-14-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-14.png" data-orig-size="675,255" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-14" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-14.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-14.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-14.png?w=675" alt="" class="wp-image-14244" srcset="https://terrytao.files.wordpress.com/2023/12/image-14.png 675w, https://terrytao.files.wordpress.com/2023/12/image-14.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-14.png?w=300 300w" sizes="(max-width: 675px) 100vw, 675px" /></a></figure>
The goal is now very close to the hypothesis <code>hi</code>. One could now look up the documentation for <code>Finset.range</code> to see how to unpack <code>hi</code>, but as before <code>simp</code> can do this for us. Invoking <code>simp at hi</code>, we obtain
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-15.png"><img loading="lazy" width="671" height="263" data-attachment-id="14247" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-15-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-15.png" data-orig-size="671,263" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-15" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-15.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-15.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-15.png?w=671" alt="" class="wp-image-14247" srcset="https://terrytao.files.wordpress.com/2023/12/image-15.png 671w, https://terrytao.files.wordpress.com/2023/12/image-15.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-15.png?w=300 300w" sizes="(max-width: 671px) 100vw, 671px" /></a></figure>
Now the goal and hypothesis are very close indeed. Here we can just close the goal using the <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#linarith">linarith</a></code> tactic used in the <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous tour</a>:
<figure class="wp-block-image size-large is-resized"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="304" data-attachment-id="14250" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-16-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-16.png" data-orig-size="1744,519" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-16" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=1024" alt="" class="wp-image-14250" style="width:840px;height:auto" srcset="https://terrytao.files.wordpress.com/2023/12/image-16.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/12/image-16.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-16.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-16.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-16.png 1744w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The next sorry can be resolved by similar methods, using the hypothesis <code>hD</code> applied at the variable <code>i</code>:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="290" data-attachment-id="14252" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-17-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-17.png" data-orig-size="1739,493" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-17" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=1024" alt="" class="wp-image-14252" srcset="https://terrytao.files.wordpress.com/2023/12/image-17.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-17.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-17.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-17.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-17.png 1739w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Now for the penultimate sorry. As in a previous step, we can use <code>congr</code> to remove the denominator, leaving us in this state:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-18.png"><img loading="lazy" width="764" height="249" data-attachment-id="14254" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-18-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-18.png" data-orig-size="764,249" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-18" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-18.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-18.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-18.png?w=764" alt="" class="wp-image-14254" srcset="https://terrytao.files.wordpress.com/2023/12/image-18.png 764w, https://terrytao.files.wordpress.com/2023/12/image-18.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-18.png?w=300 300w" sizes="(max-width: 764px) 100vw, 764px" /></a></figure>
This is a telescoping series identity. One could try to prove it by induction, or one could try to see if this identity is already in Mathlib. Searching for <code>Finset</code>, <code>sum</code>, and <code>sub</code> will <a href="https://leanprover-community.github.io/mathlib4_docs/search.html?sitesearch=https%3A%2F%2Fleanprover-community.github.io%2Fmathlib4_docs&q=finset+sum+sub">locate the right tool</a> (as the fifth hit), but a simpler way to proceed here is to use the <code>exact?</code> tactic we saw in the <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous tour</a>:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-19.png"><img loading="lazy" width="755" height="156" data-attachment-id="14256" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-19-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-19.png" data-orig-size="755,156" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-19" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-19.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-19.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-19.png?w=755" alt="" class="wp-image-14256" srcset="https://terrytao.files.wordpress.com/2023/12/image-19.png 755w, https://terrytao.files.wordpress.com/2023/12/image-19.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-19.png?w=300 300w" sizes="(max-width: 755px) 100vw, 755px" /></a></figure>
A brief check of the documentation for <code><a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/BigOperators/Basic.html#Finset.sum_range_sub'">sum_range_sub'</a></code> confirms that this is what we want. Actually we can just use <code>apply sum_range_sub'</code> here, as the <code><a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#apply">apply</a></code> tactic is smart enough to fill in the missing arguments:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20apply%20sum_range_sub'%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20sorry"><img loading="lazy" width="1024" height="279" data-attachment-id="14259" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-20-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-20.png" data-orig-size="1744,476" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-20" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=1024" alt="" class="wp-image-14259" srcset="https://terrytao.files.wordpress.com/2023/12/image-20.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-20.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-20.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-20.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-20.png 1744w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
One last <code>sorry</code> to go. As before, we use <code>gcongr</code> to cancel denominators, leaving us with
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/12/image-21.png"><img loading="lazy" width="662" height="217" data-attachment-id="14261" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-21-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-21.png" data-orig-size="662,217" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-21" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-21.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-21.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-21.png?w=662" alt="" class="wp-image-14261" srcset="https://terrytao.files.wordpress.com/2023/12/image-21.png 662w, https://terrytao.files.wordpress.com/2023/12/image-21.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-21.png?w=300 300w" sizes="(max-width: 662px) 100vw, 662px" /></a></figure>
This looks easy, because the hypothesis <code>hpos</code> will tell us that <code>D (k+1)</code> is nonnegative; specifically, the instance <code>hpos (k+1)</code> of that hypothesis will state exactly this. The <code>linarith</code> tactic will then resolve this goal once it is told about this particular instance:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20apply%20sum_range_sub'%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%3B%20linarith%20%5Bhpos%20(k%2B1)%5D"><img loading="lazy" width="1024" height="294" data-attachment-id="14264" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-22-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-22.png" data-orig-size="1782,513" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-22" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=1024" alt="" class="wp-image-14264" srcset="https://terrytao.files.wordpress.com/2023/12/image-22.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=1021 1021w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-22.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-22.png 1782w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
We now have a complete proof – no more yellow squiggly line in the example. There are two warnings though – there are two variables <code>i</code> and <code>hi</code> introduced in the proof that Lean’s “linter” has noticed are not actually used in the proof. So we can rename them with underscores to tell Lean that we are okay with them not being used:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%20%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%0D%0A%20%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20%20%20a%20k%20%3D%20(k%2B1)%20*%20(a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20field_simp%3B%20ring%0D%0A%20%20%20%20_%20%3D%20(%E2%88%91%20_i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20simp%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20a%20i)%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%20%3A%3D%20by%20gcongr%20with%20i%20_%3B%20exact%20hD%20i%0D%0A%20%20%20%20_%20%3D%20(D%200%20-%20D%20(k%2B1))%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20congr%3B%20apply%20sum_range_sub'%0D%0A%20%20%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%3A%3D%20by%20gcongr%3B%20linarith%20%5Bhpos%20(k%2B1)%5D"><img loading="lazy" width="1024" height="278" data-attachment-id="14266" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-23-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-23.png" data-orig-size="1763,479" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-23" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=1024" alt="" class="wp-image-14266" srcset="https://terrytao.files.wordpress.com/2023/12/image-23.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-23.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-23.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-23.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-23.png 1763w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
This is a perfectly fine proof, but upon noticing that many of the steps are similar to each other, one can do a bit of “code golf” as in the <a href="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/">previous tour</a> to compactify the proof a bit:
<figure class="wp-block-image size-large"><a href="https://live.lean-lang.org/#code=import%20Mathlib%0D%0A%0D%0Aopen%20Finset%20BigOperators%0D%0A%0D%0Aexample%20(a%20D%20%3A%20%E2%84%95%20%E2%86%92%20%E2%84%9D)%20(hD%20%3A%20%E2%88%80%20k%2C%20a%20k%20%E2%89%A4%20D%20k%20-%20D%20(k%2B1))%20(hpos%3A%20%E2%88%80%20k%2C%200%20%E2%89%A4%20D%20k)%20(ha%20%3A%20Antitone%20a)%20(k%20%3A%20%E2%84%95)%20%3A%20a%20k%20%E2%89%A4%20D%200%20%2F%20(k%20%2B%201)%20%3A%3D%20calc%0D%0A%20%20a%20k%20%3D%20(%E2%88%91%20_i%20in%20range%20(k%2B1)%2C%20a%20k)%20%2F%20(k%2B1)%20%3A%3D%20by%20simp%3B%20field_simp%3B%20ring%0D%0A%20%20_%20%E2%89%A4%20(%E2%88%91%20i%20in%20range%20(k%2B1)%2C%20(D%20i%20-%20D%20(i%2B1)))%20%2F%20(k%2B1)%20%3A%3D%20by%20gcongr%20with%20i%20hi%3B%20apply%20le_trans%20_%20(hD%20i)%3B%20apply%20ha%3B%20simp%20at%20hi%3B%20linarith%0D%0A%20%20_%20%E2%89%A4%20D%200%20%2F%20(k%2B1)%20%3A%3D%20by%20gcongr%3B%20rw%20%5Bsum_range_sub'%5D%3B%20linarith%20%5Bhpos%20(k%2B1)%5D"><img loading="lazy" width="1024" height="250" data-attachment-id="14269" data-permalink="https://terrytao.wordpress.com/2023/12/05/a-slightly-longer-lean-4-proof-tour/image-25-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/12/image-25.png" data-orig-size="1642,401" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-25" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=490" src="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=1024" alt="" class="wp-image-14269" srcset="https://terrytao.files.wordpress.com/2023/12/image-25.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/12/image-25.png?w=150 150w, https://terrytao.files.wordpress.com/2023/12/image-25.png?w=300 300w, https://terrytao.files.wordpress.com/2023/12/image-25.png?w=768 768w, https://terrytao.files.wordpress.com/2023/12/image-25.png 1642w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
With enough familiarity with the Lean language, this proof actually tracks quite closely with (an optimized version of) the human blueprint.
This concludes the tour of a lengthier Lean proving exercise. I am finding the pre-planning step of the proof (using an informal “blueprint” to break the proof down into extremely granular pieces) to make the formalization process significantly easier than in the past (when I often adopted a sequential process of writing one line of code at a time without first sketching out a skeleton of the argument). (The proof here took only about 15 minutes to create initially, although for this blog post I had to recreate it with screenshots and supporting links, which took significantly more time.) I believe that a realistic near-term goal for AI is to be able to fill in automatically a significant fraction of the sorts of atomic “<code>sorry</code>“s of the size one saw in this proof, allowing one to convert a blueprint to a formal Lean proof even more rapidly.
One final remark: in this tour I filled in the “<code>sorry</code>“s in the order in which they appeared, but there is actually no requirement that one does this, and once one has used a blueprint to atomize a proof into self-contained smaller pieces, one can fill them in in any order. Importantly for a group project, these micro-tasks can be parallelized, with different contributors claiming whichever “<code>sorry</code>” they feel they are qualified to solve, and working independently of each other. (And, because Lean can automatically verify if their proof is correct, there is no need to have a pre-existing bond of trust with these contributors in order to accept their contributions.) Furthermore, because the specification of a “<code>sorry</code>” someone can make a meaningful contribution to the proof by working on an extremely localized component of it without needing the mathematical expertise to understand the global argument. This is not particularly important in this simple case, where the entire lemma is not too hard to understand to a trained mathematician, but can become quite relevant for complex formalization projects.
]]></content:encoded>
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<item>
<title>Formalizing the proof of PFR in Lean4 using Blueprint: a short tour</title>
<link>https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/</link>
<comments>https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Sat, 18 Nov 2023 23:45:56 +0000</pubDate>
<category><![CDATA[expository]]></category>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[Blueprint]]></category>
<category><![CDATA[Lean4]]></category>
<category><![CDATA[Polynomial Freiman-Ruzsa conjecture]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14054</guid>
<description><![CDATA[Since the release of my preprint with Tim, Ben, and Freddie proving the Polynomial Freiman-Ruzsa (PFR) conjecture over , I (together with Yael Dillies and Bhavik Mehta) have started a collaborative project to formalize this argument in the proof assistant language Lean4. It has been less than a week since the project was launched, but […]]]></description>
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Since the release of <a href="https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/">my preprint with Tim, Ben, and Freddie</a> proving the Polynomial Freiman-Ruzsa (PFR) conjecture over <img src="https://s0.wp.com/latex.php?latex=%7B%5Cmathbb+F%7D_2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cmathbb+F%7D_2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cmathbb+F%7D_2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="{\mathbb F}_2" class="latex" />, I (together with <a href="https://github.com/YaelDillies">Yael Dillies</a> and <a href="https://github.com/b-mehta">Bhavik Mehta</a>) have started a <a href="https://teorth.github.io/pfr/">collaborative project</a> to formalize this argument in the proof assistant language <a href="https://en.wikipedia.org/wiki/Lean_(proof_assistant)">Lean4</a>. It has been less than a week since the project was launched, but it is proceeding quite well, with a significant fraction of the paper already either fully or partially formalized. The project has been greatly assisted by the <a href="https://github.com/PatrickMassot/leanblueprint">Blueprint tool</a> of <a href="https://www.imo.universite-paris-saclay.fr/~patrick.massot/en/">Patrick Massot</a>, which allows one to write a human-readable “blueprint” of the proof that is linked to the Lean formalization; similar blueprints have been <a href="https://leanprover-community.github.io/liquid/">used for other projects</a>, such as Scholze’s <a href="https://github.com/leanprover-community/lean-liquid">liquid tensor experiment</a>. For the PFR project, the blueprint can be found <a href="https://teorth.github.io/pfr/blueprint/">here</a>. One feature of the blueprint that I find particularly appealing is the dependency graph that is automatically generated from the blueprint, and can provide a rough snapshot of how far along the formalization has advanced. For PFR, the latest state of the dependency graph can be found <a href="https://teorth.github.io/pfr/blueprint/dep_graph_document.html">here</a>. At the current time of writing, the graph looks like this:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image.png"><img loading="lazy" width="1024" height="367" data-attachment-id="14060" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image.png" data-orig-size="2692,965" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image.png?w=1024" alt="" class="wp-image-14060" srcset="https://terrytao.files.wordpress.com/2023/11/image.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image.png?w=2048 2048w, https://terrytao.files.wordpress.com/2023/11/image.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image.png?w=768 768w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The color coding of the various bubbles (for lemmas) and rectangles (for definitions) is explained in the legend to <a href="https://teorth.github.io/pfr/blueprint/dep_graph_document.html">the dependency graph</a>, but roughly speaking the green bubbles/rectangles represent lemmas or definitions that have been fully formalized, and the blue ones represent lemmas or definitions which are ready to be formalized (their statements, but not proofs, have already been formalized, as well as those of all prerequisite lemmas and proofs). The goal is to get all the bubbles leading up to and including the “<a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">pfr</a>” bubble at the bottom colored in green.
In this post I would like to give a quick “tour” of the project, to give a sense of how it operates. If one clicks on the “<a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">pfr</a>” bubble at the bottom of the dependency graph, we get the following:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-1.png"><img loading="lazy" width="1024" height="265" data-attachment-id="14064" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-1/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-1.png" data-orig-size="2390,620" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-1" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=1024" alt="" class="wp-image-14064" srcset="https://terrytao.files.wordpress.com/2023/11/image-1.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=2043 2043w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-1.png?w=768 768w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Here, Blueprint is displaying a human-readable form of the PFR statement. This is coming from the <a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">corresponding portion of the blueprint</a>, which also comes with a human-readable proof of this statement that relies on other statements in the project:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-2.png"><img loading="lazy" width="1024" height="449" data-attachment-id="14066" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-2/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-2.png" data-orig-size="1744,765" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-2" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=1024" alt="" class="wp-image-14066" srcset="https://terrytao.files.wordpress.com/2023/11/image-2.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-2.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-2.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-2.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-2.png 1744w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
(I have cropped out the second half of the proof here, as it is not relevant to the discussion.)
Observe that the “<a href="https://teorth.github.io/pfr/blueprint/sect0007.html#pfr">pfr</a>” bubble is white, but has a green border. This means that the statement of PFR has been formalized in Lean, but not the proof; and the proof itself is not ready to be formalized, because some of the prerequisites (in particular, “<a href="https://teorth.github.io/pfr/blueprint/sect0006.html#entropy-pfr">entropy-pfr</a>” (Theorem 6.16)) do not even have their statements formalized yet. If we click on the “Lean” link below the description of PFR in the dependency graph, we are lead to the (auto-generated) <a href="https://teorth.github.io/pfr/docs/PFR/main.html#PFR_conjecture">Lean documentation for this assertion</a>:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-3.png"><img loading="lazy" width="1024" height="281" data-attachment-id="14068" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-3/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-3.png" data-orig-size="1533,421" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-3" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=1024" alt="" class="wp-image-14068" srcset="https://terrytao.files.wordpress.com/2023/11/image-3.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-3.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-3.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-3.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-3.png 1533w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
This is what a typical theorem in Lean looks like (after a procedure known as “pretty printing”). There are a number of hypotheses stated before the colon, for instance that <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> is a finite elementary abelian group of order <img src="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2" class="latex" /> (this is how we have chosen to formalize the finite field vector spaces <img src="https://s0.wp.com/latex.php?latex=%7B%5Cbf+F%7D_2%5En&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cbf+F%7D_2%5En&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cbf+F%7D_2%5En&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="{\bf F}_2^n" class="latex" />), that <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" /> is a non-empty subset of <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> (the hypothesis that <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" /> is non-empty was not stated in the LaTeX version of the conjecture, but we realized it was necessary in the formalization, and will update the LaTeX blueprint shortly to reflect this) with the cardinality of <img src="https://s0.wp.com/latex.php?latex=A%2BA&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A%2BA&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A%2BA&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A+A" class="latex" /> less than <img src="https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="K" class="latex" /> times the cardinality of <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" />, and the statement after the colon is the conclusion: that <img src="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" /> can be contained in the sum <img src="https://s0.wp.com/latex.php?latex=c%2BH&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c%2BH&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c%2BH&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="c+H" class="latex" /> of a subgroup <img src="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> and a set <img src="https://s0.wp.com/latex.php?latex=c&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="c" class="latex" /> of cardinality at most <img src="https://s0.wp.com/latex.php?latex=2K%5E%7B12%7D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2K%5E%7B12%7D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2K%5E%7B12%7D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2K^{12}" class="latex" />.
The astute reader may notice that the above theorem seems to be missing one or two details, for instance it does not explicitly assert that <img src="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H" class="latex" /> is a subgroup. This is because the “pretty printing” suppresses some of the information in the actual statement of the theorem, which can be seen by clicking on the “<a href="https://github.com/teorth/pfr/blob/6b2c357dc922ff00dec5dd6873b841925b0e22ef//PFR/main.lean#L17-L21">Source</a>” link:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-4.png"><img loading="lazy" width="1024" height="174" data-attachment-id="14072" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-4/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-4.png" data-orig-size="1737,296" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-4" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=1024" alt="" class="wp-image-14072" srcset="https://terrytao.files.wordpress.com/2023/11/image-4.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-4.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-4.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-4.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-4.png 1737w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Here we see that <img src="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H" class="latex" /> is required to have the “type” of an additive subgroup of <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" />. (Lean’s language revolves very strongly around <a href="https://en.wikipedia.org/wiki/Type_theory">types</a>, but for this tour we will not go into detail into what a type is exactly.) The prominent “sorry” at the bottom of this theorem asserts that a proof is not yet provided for this theorem, but the intention of course is to replace this “sorry” with an actual proof eventually.
Filling in this “sorry” is too hard to do right now, so let’s look for a simpler task to accomplish instead. Here is a simple intermediate lemma “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>” that shows up in the proof:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-5.png"><img loading="lazy" width="1024" height="371" data-attachment-id="14075" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-5/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-5.png" data-orig-size="1647,597" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-5" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=1024" alt="" class="wp-image-14075" srcset="https://terrytao.files.wordpress.com/2023/11/image-5.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-5.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-5.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-5.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-5.png 1647w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The expression <img src="https://s0.wp.com/latex.php?latex=d%5BX%3B+Y%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=d%5BX%3B+Y%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=d%5BX%3B+Y%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="d[X; Y]" class="latex" /> refers to something called the entropic Ruzsa distance between <img src="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=Y&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Y&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Y&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="Y" class="latex" />, which is something that is defined <a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruz-dist-def">elsewhere in the project</a>, but for the current discussion it is not important to know its precise definition, other than that it is a real number. The bubble is blue with a green border, which means that the statement has been formalized, and the proof is ready to be formalized also. The blueprint dependency graph indicates that this lemma can be deduced from just one preceding lemma, called “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-diff">ruzsa-diff</a>“:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-6.png"><img loading="lazy" width="1024" height="333" data-attachment-id="14078" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-6/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-6.png" data-orig-size="1716,559" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-6" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=1024" alt="" class="wp-image-14078" srcset="https://terrytao.files.wordpress.com/2023/11/image-6.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/11/image-6.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-6.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-6.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-6.png 1716w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
“<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-diff">ruzsa-diff</a>” is also blue and bordered in green, so it has the same current status as “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>“: the statement is formalized, and the proof is ready to be formalized also, but the proof has not been written in Lean yet. The quantity <img src="https://s0.wp.com/latex.php?latex=H%5BX%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H%5BX%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H%5BX%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="H[X]" class="latex" />, by the way, refers to the <a href="https://en.wikipedia.org/wiki/Entropy_(information_theory)">Shannon entropy</a> of <img src="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X" class="latex" />, defined <a href="https://teorth.github.io/pfr/blueprint/sect0002.html#entropy-def">elsewhere in the project</a>, but for this discussion we do not need to know its definition, other than to know that it is a real number.
Looking at Lemma 3.11 and Lemma 3.13 it is clear how the former will imply the latter: the quantity <img src="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="|H[X] - H[Y]|" class="latex" /> is clearly non-negative! (There is a factor of <img src="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2" class="latex" /> present in Lemma 3.11, but it can be easily canceled out.) So it should be an easy task to fill in the proof of Lemma 3.13 assuming Lemma 3.11, even if we still don’t know how to prove Lemma 3.11 yet. Let’s first look at the Lean code for each lemma. Lemma 3.11 is formalized <a href="https://github.com/teorth/pfr/blob/6b2c357dc922ff00dec5dd6873b841925b0e22ef//PFR/ruzsa_distance.lean#L118-L118">as follows</a>:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-7.png"><img loading="lazy" width="1024" height="80" data-attachment-id="14082" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-7/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-7.png" data-orig-size="1185,93" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-7" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=1024" alt="" class="wp-image-14082" srcset="https://terrytao.files.wordpress.com/2023/11/image-7.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-7.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-7.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-7.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-7.png 1185w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Again we have a “sorry” to indicate that this lemma does not currently have a proof. The Lean notation (as well as the name of the lemma) differs a little from the LaTeX version for technical reasons that we will not go into here. (Also, the variables <img src="https://s0.wp.com/latex.php?latex=X%2C+%5Cmu%2C+Y%2C+%5Cmu%27&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X%2C+%5Cmu%2C+Y%2C+%5Cmu%27&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X%2C+%5Cmu%2C+Y%2C+%5Cmu%27&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X, \mu, Y, \mu'" class="latex" /> are introduced at an earlier stage in the Lean file; again, we will ignore this point for the ensuing discussion.) Meanwhile, Lemma 3.13 is <a href="https://github.com/teorth/pfr/blob/6b2c357dc922ff00dec5dd6873b841925b0e22ef//PFR/ruzsa_distance.lean#L127-L128">currently formalized as</a>
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-8.png"><img loading="lazy" width="822" height="108" data-attachment-id="14085" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-8/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-8.png" data-orig-size="822,108" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-8" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-8.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-8.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-8.png?w=822" alt="" class="wp-image-14085" srcset="https://terrytao.files.wordpress.com/2023/11/image-8.png 822w, https://terrytao.files.wordpress.com/2023/11/image-8.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-8.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-8.png?w=768 768w" sizes="(max-width: 822px) 100vw, 822px" /></a></figure>
OK, I’m now going to try to fill in the latter “sorry”. In my local copy of the <a href="https://github.com/teorth/pfr">PFR github repository</a>, I open up the <a href="https://github.com/teorth/pfr/blob/master/PFR/ruzsa_distance.lean">relevant Lean file</a> in my editor (<a href="https://code.visualstudio.com/">Visual Studio Code</a>, with the <a href="https://github.com/leanprover/vscode-lean4">lean4 extension</a>) and navigate to the “sorry” of “rdist_nonneg”. The accompanying “Lean infoview” then shows the current state of the Lean proof:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-9.png"><img loading="lazy" width="605" height="1024" data-attachment-id="14087" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-9/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-9.png" data-orig-size="614,1040" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-9" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=177" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=605" alt="" class="wp-image-14087" srcset="https://terrytao.files.wordpress.com/2023/11/image-9.png?w=605 605w, https://terrytao.files.wordpress.com/2023/11/image-9.png?w=89 89w, https://terrytao.files.wordpress.com/2023/11/image-9.png?w=177 177w, https://terrytao.files.wordpress.com/2023/11/image-9.png 614w" sizes="(max-width: 605px) 100vw, 605px" /></a></figure>
Here we see a number of ambient hypotheses (e.g., that <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" /> is an additive commutative group, that <img src="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X" class="latex" /> is a map from <img src="https://s0.wp.com/latex.php?latex=%5COmega&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5COmega&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5COmega&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\Omega" class="latex" /> to <img src="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="G" class="latex" />, and so forth; many of these hypotheses are not actually relevant for this particular lemma), and at the bottom we see the goal we wish to prove.
OK, so now I’ll try to prove the claim. This is accomplished by applying a series of “tactics” to transform the goal and/or hypotheses. The first step I’ll do is to put in the factor of <img src="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="2" class="latex" /> that is needed to apply Lemma 3.11. This I will do with the “suffices” tactic, writing in the proof
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-10.png"><img loading="lazy" width="812" height="204" data-attachment-id="14090" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-10/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-10.png" data-orig-size="812,204" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-10" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-10.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-10.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-10.png?w=812" alt="" class="wp-image-14090" srcset="https://terrytao.files.wordpress.com/2023/11/image-10.png 812w, https://terrytao.files.wordpress.com/2023/11/image-10.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-10.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-10.png?w=768 768w" sizes="(max-width: 812px) 100vw, 812px" /></a></figure>
I now have two goals (and two “sorries”): one to show that <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" /> implies <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+d%5BX%2CY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+d%5BX%2CY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+d%5BX%2CY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq d[X,Y]" class="latex" />, and the other to show that <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" />. (The yellow squiggly underline indicates that this lemma has not been fully proven yet due to the presence of “sorry”s. The dot “.” is a syntactic marker that is useful to separate the two goals from each other, but you can ignore it for this tour.) The Lean tactic “suffices” corresponds, roughly speaking, to the phrase “It suffices to show that …” (or more precisely, “It suffices to show that … . To see this, … . It remains to verify the claim …”) in Mathematical English. For my own education, I wrote a “Lean phrasebook” of further correspondences between lines of Lean code and sentences or phrases in Mathematical English, which can be found <a href="https://docs.google.com/spreadsheets/d/1Gsn5al4hlpNc_xKoXdU6XGmMyLiX4q-LFesFVsMlANo/edit#gid=0">here</a>.
Let’s fill in the first “sorry”. The tactic state now looks like this (cropping out some irrelevant hypotheses):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-11.png"><img loading="lazy" width="532" height="206" data-attachment-id="14093" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-11/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-11.png" data-orig-size="532,206" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-11" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-11.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-11.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-11.png?w=532" alt="" class="wp-image-14093" srcset="https://terrytao.files.wordpress.com/2023/11/image-11.png 532w, https://terrytao.files.wordpress.com/2023/11/image-11.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-11.png?w=300 300w" sizes="(max-width: 532px) 100vw, 532px" /></a></figure>
Here I can use a handy tactic “<a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#linarith">linarith</a>“, which solves any goal that can be derived by linear arithmetic from existing hypotheses:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-12.png"><img loading="lazy" width="829" height="218" data-attachment-id="14095" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-12/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-12.png" data-orig-size="829,218" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-12" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-12.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-12.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-12.png?w=829" alt="" class="wp-image-14095" srcset="https://terrytao.files.wordpress.com/2023/11/image-12.png 829w, https://terrytao.files.wordpress.com/2023/11/image-12.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-12.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-12.png?w=768 768w" sizes="(max-width: 829px) 100vw, 829px" /></a></figure>
This works, and now the tactic state reports no goals left to prove on this branch, so we move on to the remaining sorry, in which the goal is now to prove <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" />:
<figure class="wp-block-image size-large is-resized"><a href="https://terrytao.files.wordpress.com/2023/11/image-13.png"><img loading="lazy" width="507" height="177" data-attachment-id="14097" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-13/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-13.png" data-orig-size="507,177" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-13" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-13.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-13.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-13.png?w=507" alt="" class="wp-image-14097" style="width:840px;height:auto" srcset="https://terrytao.files.wordpress.com/2023/11/image-13.png 507w, https://terrytao.files.wordpress.com/2023/11/image-13.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-13.png?w=300 300w" sizes="(max-width: 507px) 100vw, 507px" /></a></figure>
Here we will try to invoke Lemma 3.11. I add the following lines of code:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-14.png"><img loading="lazy" width="1024" height="301" data-attachment-id="14099" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-14/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-14.png" data-orig-size="1042,307" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-14" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=1024" alt="" class="wp-image-14099" srcset="https://terrytao.files.wordpress.com/2023/11/image-14.png?w=1022 1022w, https://terrytao.files.wordpress.com/2023/11/image-14.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-14.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-14.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-14.png 1042w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
The Lean tactic “<a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#have">have</a>” roughly corresponds to the Mathematical English phrase “We have the statement…” or “We claim the statement…”; like “suffices”, it splits a goal into two subgoals, though in the reversed order to “suffices”.
I again have two subgoals, one to prove the bound <img src="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D-H%5BY%5D%7C+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7CH%5BX%5D-H%5BY%5D%7C+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7CH%5BX%5D-H%5BY%5D%7C+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="|H[X]-H[Y]| \leq 2 d[X;Y]" class="latex" /> (which I will call “h”), and then to deduce the previous goal <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+2+d%5BX%3BY%5D&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq 2 d[X;Y]" class="latex" /> from <img src="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="h" class="latex" />. For the first, I know I should invoke the lemma “diff_ent_le_rdist” that is encoding Lemma 3.11. One way to do this is to try the tactic “exact?”, which will automatically search to see if the goal can already be deduced immediately from an existing lemma. It reports:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-15.png"><img loading="lazy" width="553" height="127" data-attachment-id="14101" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-15/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-15.png" data-orig-size="553,127" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-15" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-15.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-15.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-15.png?w=553" alt="" class="wp-image-14101" srcset="https://terrytao.files.wordpress.com/2023/11/image-15.png 553w, https://terrytao.files.wordpress.com/2023/11/image-15.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-15.png?w=300 300w" sizes="(max-width: 553px) 100vw, 553px" /></a></figure>
So I try this (by clicking on the suggested code, which automatically pastes it into the right location), and it works, leaving me with the final “sorry”:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-16.png"><img loading="lazy" width="1020" height="279" data-attachment-id="14103" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-16/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-16.png" data-orig-size="1020,279" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-16" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-16.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-16.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-16.png?w=1020" alt="" class="wp-image-14103" srcset="https://terrytao.files.wordpress.com/2023/11/image-16.png 1020w, https://terrytao.files.wordpress.com/2023/11/image-16.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-16.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-16.png?w=768 768w" sizes="(max-width: 1020px) 100vw, 1020px" /></a></figure>
The lean tactic “<a href="https://leanprover-community.github.io/mathlib_docs/tactics.html#exact">exact</a>” corresponds, roughly speaking, to the Mathematical English phrase “But this is exactly …”.
At this point I should mention that I also have the <a href="https://github.com/features/copilot">Github Copilot</a> extension to Visual Studio Code installed. This is an AI which acts as an advanced autocomplete that can suggest possible lines of code as one types. In this case, it offered a suggestion which was almost correct (the second line is what we need, whereas the first is not necessary, and in fact does not even compile in Lean):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-25.png"><img loading="lazy" width="1024" height="276" data-attachment-id="14135" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-25/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-25.png" data-orig-size="1193,322" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-25" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=1024" alt="" class="wp-image-14135" srcset="https://terrytao.files.wordpress.com/2023/11/image-25.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-25.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-25.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-25.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-25.png 1193w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
In any event, “exact?” worked in this case, so I can ignore the suggestion of Copilot this time (it has been very useful in other cases though). I apply the “exact?” tactic a second time and follow its suggestion to establish the matching bound <img src="https://s0.wp.com/latex.php?latex=0+%5Cleq+%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=0+%5Cleq+%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=0+%5Cleq+%7CH%5BX%5D+-+H%5BY%5D%7C&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="0 \leq |H[X] - H[Y]|" class="latex" />:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-17.png"><img loading="lazy" width="1013" height="354" data-attachment-id="14105" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-17/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-17.png" data-orig-size="1013,354" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-17" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-17.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-17.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-17.png?w=1013" alt="" class="wp-image-14105" srcset="https://terrytao.files.wordpress.com/2023/11/image-17.png 1013w, https://terrytao.files.wordpress.com/2023/11/image-17.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-17.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-17.png?w=768 768w" sizes="(max-width: 1013px) 100vw, 1013px" /></a></figure>
(One can find documention for the “abs_nonneg” method <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Order/Group/Abs.html#abs_nonneg">here</a>. Copilot, by the way, was also able to resolve this step, albeit with a slightly different syntax; there are also several other search engines available to locate this method as well, such as <a href="https://moogle-morphlabs.vercel.app/search/raw?q=absolute%20value%20is%20nonnegative">Moogle</a>. One of the main purposes of the <a href="https://leanprover-community.github.io/contribute/naming.html">Lean naming conventions for lemmas</a>, by the way, is to facilitate the location of methods such as “abs_nonneg”, which is easier figure out how to search for than a method named (say) “Lemma 1.2.1”.) To fill in the final “sorry”, I try “exact?” one last time, to figure out how to combine <img src="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="h" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=h%27&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=h%27&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=h%27&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="h'" class="latex" /> to give the desired goal, and it works!
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-18.png"><img loading="lazy" width="1024" height="362" data-attachment-id="14107" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-18/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-18.png" data-orig-size="1027,364" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-18" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=1024" alt="" class="wp-image-14107" srcset="https://terrytao.files.wordpress.com/2023/11/image-18.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=1021 1021w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-18.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-18.png 1027w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
Note that all the squiggly underlines have disappeared, indicating that Lean has accepted this as a valid proof. The documentation for “ge_trans” may be found <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/Defs.html#ge_trans">here</a>. The reader may observe that this method uses the <img src="https://s0.wp.com/latex.php?latex=%5Cgeq&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cgeq&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cgeq&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\geq" class="latex" /> relation rather than the <img src="https://s0.wp.com/latex.php?latex=%5Cleq&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleq&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleq&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="\leq" class="latex" /> relation, but in Lean the assertions <img src="https://s0.wp.com/latex.php?latex=X+%5Cgeq+Y&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X+%5Cgeq+Y&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X+%5Cgeq+Y&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X \geq Y" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=Y+%5Cleq+X&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Y+%5Cleq+X&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Y+%5Cleq+X&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="Y \leq X" class="latex" /> are “<a href="https://www.ma.imperial.ac.uk/~buzzard/xena/formalising-mathematics-2022/Part_B/equality.html#:~:text=Definitional%20equality,-Definitional%20equality%20is&text=In%20Lean%2C%20%C2%ACP%20is,equal%2C%20they%20are%20definitionally%20equal.">definitionally equal</a>“, allowing tactics such as “exact” to use them interchangeably. “exact <a href="https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Order/Defs.html#le_trans">le_trans</a> h’ h” would also have worked in this instance.
It is possible to compactify this proof quite a bit by cutting out several intermediate steps (a procedure sometimes known as “<a href="https://en.wikipedia.org/wiki/Code_golf">code golf</a>“):
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-19.png"><img loading="lazy" width="1024" height="130" data-attachment-id="14109" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-19/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-19.png" data-orig-size="1190,152" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-19" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=1024" alt="" class="wp-image-14109" srcset="https://terrytao.files.wordpress.com/2023/11/image-19.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=1018 1018w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-19.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-19.png 1190w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
And now the proof is done! In the end, it was literally a “one-line proof”, which makes sense given how close Lemma 3.11 and Lemma 3.13 were to each other.
The current version of Blueprint does not automatically verify the proof (even though it does compile in Lean), so we have to manually update the blueprint as well. The LaTeX for Lemma 3.13 <a href="https://github.com/teorth/pfr/blob/master/blueprint/src/chapter/distance.tex">currently looks like this:</a>
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-20.png"><img loading="lazy" width="1024" height="309" data-attachment-id="14111" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-20/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-20.png" data-orig-size="1130,341" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-20" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=1024" alt="" class="wp-image-14111" srcset="https://terrytao.files.wordpress.com/2023/11/image-20.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-20.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-20.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-20.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-20.png 1130w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
I add the “\leanok” macro to the proof, to flag that the proof has now been formalized:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-22.png"><img loading="lazy" width="1024" height="284" data-attachment-id="14114" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-22/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-22.png" data-orig-size="1217,338" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-22" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=1024" alt="" class="wp-image-14114" srcset="https://terrytao.files.wordpress.com/2023/11/image-22.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-22.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-22.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-22.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-22.png 1217w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
I then push everything back up to the master Github repository. The blueprint will take quite some time (about half an hour) to rebuild, but eventually it does, and the dependency graph (which Blueprint has for some reason decided to rearrange a bit) now shows “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>” in green:
<figure class="wp-block-image size-large"><a href="https://terrytao.files.wordpress.com/2023/11/image-23.png"><img loading="lazy" width="1024" height="443" data-attachment-id="14121" data-permalink="https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/image-23/" data-orig-file="https://terrytao.files.wordpress.com/2023/11/image-23.png" data-orig-size="1643,711" data-comments-opened="1" data-image-meta="{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}" data-image-title="image-23" data-image-description="" data-image-caption="" data-medium-file="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=300" data-large-file="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=490" src="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=1024" alt="" class="wp-image-14121" srcset="https://terrytao.files.wordpress.com/2023/11/image-23.png?w=1024 1024w, https://terrytao.files.wordpress.com/2023/11/image-23.png?w=150 150w, https://terrytao.files.wordpress.com/2023/11/image-23.png?w=300 300w, https://terrytao.files.wordpress.com/2023/11/image-23.png?w=768 768w, https://terrytao.files.wordpress.com/2023/11/image-23.png 1643w" sizes="(max-width: 1024px) 100vw, 1024px" /></a></figure>
And so the formalization of PFR moves a little bit closer to completion. (Of course, this was a particularly easy lemma to formalize, that I chose to illustrate the process; one can imagine that most other lemmas will take a bit more work.) Note that while “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-nonneg">ruzsa-nonneg</a>” is now colored in green, we don’t yet have a full proof of this result, because the lemma “<a href="https://teorth.github.io/pfr/blueprint/sect0003.html#ruzsa-diff">ruzsa-diff</a>” that it relies on is not green. Nevertheless, the proof is locally complete at this point; hopefully at some point in the future, the predecessor results will also be locally proven, at which point this result will be completely proven. Note how this blueprint structure allows one to work on different parts of the proof asynchronously; it is not necessary to wait for earlier stages of the argument to be fully formalized to start working on later stages, although I anticipate a small amount of interaction between different components as we iron out any bugs or slight inaccuracies in the blueprint. (For instance, I am suspecting that we may need to add some measurability hypotheses on the random variables <img src="https://s0.wp.com/latex.php?latex=X%2C+Y&bg=ffffff&fg=545454&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=X%2C+Y&bg=ffffff&fg=545454&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=X%2C+Y&bg=ffffff&fg=545454&s=0&c=20201002&zoom=4.5 4x" alt="X, Y" class="latex" /> in the above two lemmas to make them completely true, but this is something that should emerge organically as the formalization process continues.)
That concludes the brief tour! If you are interested in learning more about the project, you can follow the <a href="https://leanprover.zulipchat.com/#narrow/stream/412902-Polynomial-Freiman-Ruzsa-conjecture">Zulip chat stream</a>; you can also <a href="https://leanprover-community.github.io/get_started.html">download Lean</a> and <a href="https://leanprover-community.github.io/install/project.html">work on the PFR project yourself</a>, using a local copy of the Github repository and sending pull requests to the master copy if you have managed to fill in one or more of the “sorry”s in the current version (but if you plan to work on anything more large scale than filling in a small lemma, it is good to announce your intention on the Zulip chat to avoid duplication of effort) . (One key advantage of working with a project based around a proof assistant language such as Lean is that it makes large-scale mathematical collaboration possible without necessarily having a pre-established level of trust amongst the collaborators; my fellow repository maintainers and I have already approved several pull requests from contributors that had not previously met, as the code was verified to be correct and we could see that it advanced the project. Conversely, as the above example should hopefully demonstrate, it is possible for a contributor to work on one small corner of the project without necessarily needing to understand all the mathematics that goes into the project as a whole.)
If one just wants to experiment with Lean without going to the effort of downloading it, you can playing try the “<a href="https://adam.math.hhu.de/">Natural Number Game</a>” for a gentle introduction to the language, or the <a href="https://live.lean-lang.org/">Lean4 playground</a> for an online Lean server. Further resources to learn Lean4 may be found <a href="https://leanprover-community.github.io/learn.html">here</a>.
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<title>On a conjecture of Marton</title>
<link>https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/</link>
<comments>https://terrytao.wordpress.com/2023/11/13/on-a-conjecture-of-marton/#comments</comments>
<dc:creator><![CDATA[Terence Tao]]></dc:creator>
<pubDate>Mon, 13 Nov 2023 17:41:05 +0000</pubDate>
<category><![CDATA[math.CO]]></category>
<category><![CDATA[paper]]></category>
<category><![CDATA[additive combinatorics]]></category>
<category><![CDATA[Ben Green]]></category>
<category><![CDATA[Freddie Manners]]></category>
<category><![CDATA[Freiman's theorem]]></category>
<category><![CDATA[Polynomial Freiman-Ruzsa conjecture]]></category>
<category><![CDATA[Shannon entropy]]></category>
<guid isPermaLink="false">http://terrytao.wordpress.com/?p=14046</guid>
<description><![CDATA[Tim Gowers, Ben Green, Freddie Manners, and I have just uploaded to the arXiv our paper “On a conjecture of Marton“. This paper establishes a version of the notorious Polynomial Freiman–Ruzsa conjecture (first proposed by Katalin Marton): Theorem 1 (Polynomial Freiman–Ruzsa conjecture) Let be such that . Then can be covered by at most translates […]]]></description>
<content:encoded><![CDATA[

<a href="https://www.dpmms.cam.ac.uk/~wtg10/">Tim Gowers</a>, <a href="https://people.maths.ox.ac.uk/greenbj/">Ben Green</a>, <a href="https://mathweb.ucsd.edu/~fmanners/">Freddie Manners</a>, and I have just uploaded to the arXiv our paper “<a href="https://arxiv.org/abs/2311.05762">On a conjecture of Marton</a>“. This paper establishes a version of the notorious <a href="https://terrytao.wordpress.com/2007/03/11/ben-green-the-polynomial-freiman-ruzsa-conjecture/">Polynomial Freiman–Ruzsa conjecture</a> (first proposed by Katalin Marton):

<blockquote>Theorem 1 (Polynomial Freiman–Ruzsa conjecture) Let <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Csubset+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \subset {\bf F}_2^n}" class="latex" /> be such that <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C+%5Cleq+K%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A+A| \leq K|A|}" class="latex" />. Then <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> can be covered by at most <img src="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2K^{12}}" class="latex" /> translates of a subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" /> of cardinality at most <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" />. </blockquote>


The previous best known result towards this conjecture was by Konyagin (as communicated in <a HREF="https://zbmath.org/1337.11014">this paper of Sanders</a>), who obtained a similar result but with <img src="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2K%5E%7B12%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2K^{12}}" class="latex" /> replaced by <img src="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28O_%5Cvarepsilon%28%5Clog%5E%7B3%2B%5Cvarepsilon%7D+K%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cexp%28O_%5Cvarepsilon%28%5Clog%5E%7B3%2B%5Cvarepsilon%7D+K%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cexp%28O_%5Cvarepsilon%28%5Clog%5E%7B3%2B%5Cvarepsilon%7D+K%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\exp(O_\varepsilon(\log^{3+\varepsilon} K))}" class="latex" /> for any <img src="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cvarepsilon%3E0%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\varepsilon>0}" class="latex" /> (assuming that say <img src="https://s0.wp.com/latex.php?latex=%7BK+%5Cgeq+3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK+%5Cgeq+3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK+%5Cgeq+3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K \geq 3/2}" class="latex" /> to avoid some degeneracies as <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> approaches <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" />, which is not the difficult case of the conjecture). The conjecture (with <img src="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{12}" class="latex" /> replaced by an unspecified constant <img src="https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BC%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{C}" class="latex" />) has a number of equivalent forms; see this <a href="https://zbmath.org/1155.11306">survey of Green</a>, and these papers <a href="https://zbmath.org/1274.11158">of Lovett</a> and <a href="https://zbmath.org/1229.11132">of Green and myself</a> for some examples; in particular, as discussed in the latter two references, the constants in the inverse <img src="https://s0.wp.com/latex.php?latex=%7BU%5E3%28%7B%5Cbf+F%7D_2%5En%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BU%5E3%28%7B%5Cbf+F%7D_2%5En%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BU%5E3%28%7B%5Cbf+F%7D_2%5En%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{U^3({\bf F}_2^n)}" class="latex" /> theorem are now polynomial in nature (although we did not try to optimize the constant).

The exponent <img src="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B12%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{12}" class="latex" /> here was the product of a large number of optimizations to the argument (our original exponent here was closer to <img src="https://s0.wp.com/latex.php?latex=%7B1000%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1000%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1000%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1000}" class="latex" />), but can be improved even further with additional effort (our current argument, for instance, allows one to replace it with <img src="https://s0.wp.com/latex.php?latex=%7B7%2B%5Csqrt%7B17%7D+%3D+11.123%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B7%2B%5Csqrt%7B17%7D+%3D+11.123%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B7%2B%5Csqrt%7B17%7D+%3D+11.123%5Cdots%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{7+\sqrt{17} = 11.123\dots}" class="latex" />, but we decided to state our result using integer exponents instead).

In this paper we will focus exclusively on the characteristic <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" /> case (so we will be cavalier in identifying addition and subtraction), but in a followup paper we will establish similar results in other finite characteristics.

Much of the prior progress on this sort of result has proceeded via Fourier analysis. Perhaps surprisingly, our approach uses no Fourier analysis whatsoever, being conducted instead entirely in “physical space”. Broadly speaking, it follows a natural strategy, which is to induct on the doubling constant <img src="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C%2F%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C%2F%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7CA%2BA%7C%2F%7CA%7C%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{|A+A|/|A|}" class="latex" />. Indeed, suppose for instance that one could show that every set <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> of doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> was “commensurate” in some sense to a set <img src="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A'}" class="latex" /> of doubling constant at most <img src="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^{0.99}}" class="latex" />. One measure of commensurability, for instance, might be the Ruzsa distance <img src="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%27%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CA%27%7C%5E%7B1%2F2%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%27%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CA%27%7C%5E%7B1%2F2%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%27%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CA%27%7C%5E%7B1%2F2%7D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\log \frac{|A+A'|}{|A|^{1/2} |A'|^{1/2}}}" class="latex" />, which one might hope to control by <img src="https://s0.wp.com/latex.php?latex=%7BO%28%5Clog+K%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BO%28%5Clog+K%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BO%28%5Clog+K%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{O(\log K)}" class="latex" />. Then one could iterate this procedure until doubling constant dropped below say <img src="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{3/2}" class="latex" />, at which point the conjecture is known to hold (there is an elementary argument that if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> has doubling constant less than <img src="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B3%2F2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{3/2}" class="latex" />, then <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> is in fact a subspace of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" />). One can then use several applications of the Ruzsa triangle inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Cfrac%7B%7CA%2BC%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D+%5Cleq+%5Clog+%5Cfrac%7B%7CA%2BB%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CB%7C%5E%7B1%2F2%7D%7D+%2B+%5Clog+%5Cfrac%7B%7CB%2BC%7C%7D%7B%7CB%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Cfrac%7B%7CA%2BC%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D+%5Cleq+%5Clog+%5Cfrac%7B%7CA%2BB%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CB%7C%5E%7B1%2F2%7D%7D+%2B+%5Clog+%5Cfrac%7B%7CB%2BC%7C%7D%7B%7CB%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Clog+%5Cfrac%7B%7CA%2BC%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D+%5Cleq+%5Clog+%5Cfrac%7B%7CA%2BB%7C%7D%7B%7CA%7C%5E%7B1%2F2%7D+%7CB%7C%5E%7B1%2F2%7D%7D+%2B+%5Clog+%5Cfrac%7B%7CB%2BC%7C%7D%7B%7CB%7C%5E%7B1%2F2%7D+%7CC%7C%5E%7B1%2F2%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle \log \frac{|A+C|}{|A|^{1/2} |C|^{1/2}} \leq \log \frac{|A+B|}{|A|^{1/2} |B|^{1/2}} + \log \frac{|B+C|}{|B|^{1/2} |C|^{1/2}}" class="latex" />
to conclude (the fact that we reduce <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> to <img src="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E%7B0.99%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^{0.99}}" class="latex" /> means that the various Ruzsa distances that need to be summed are controlled by a convergent geometric series).

There are a number of possible ways to try to “improve” a set <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> of not too large doubling by replacing it with a commensurate set of better doubling. We note two particular potential improvements:

<ul> <li>(i) Replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" />. For instance, if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> was a random subset (of density <img src="https://s0.wp.com/latex.php?latex=%7B1%2FK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2FK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2FK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/K}" class="latex" />) of a large subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" />, then replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> usually drops the doubling constant from <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> down to nearly <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" /> (under reasonable choices of parameters). </li><li>(ii) Replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> for a “typical” <img src="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bh+%5Cin+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{h \in A+A}" class="latex" />. For instance, if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> was the union of <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> random cosets of a subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" /> of large codimension, then replacing <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> again usually drops the doubling constant from <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> down to nearly <img src="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1}" class="latex" />.
</li></ul>

Unfortunately, there are sets <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> where neither of the above two operations (i), (ii) significantly improves the doubling constant. For instance, if <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> is a random density <img src="https://s0.wp.com/latex.php?latex=%7B1%2F%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B1%2F%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B1%2F%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{1/\sqrt{K}}" class="latex" /> subset of <img src="https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Csqrt%7BK%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\sqrt{K}}" class="latex" /> random translates of a medium-sized subspace <img src="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BH%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{H}" class="latex" />, one can check that the doubling constant stays close to <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" /> if one applies either operation (i) or operation (ii). But in this case these operations don’t actually worsen the doubling constant much either, and by applying some combination of (i) and (ii) (either intersecting <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> with a translate, or taking a sumset of <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> with itself) one can start lowering the doubling constant again.

This begins to suggest a potential strategy: show that at least one of the operations (i) or (ii) will improve the doubling constant, or at least not worsen it too much; and in the latter case, perform some more complicated operation to locate the desired doubling constant improvement.

A sign that this strategy might have a chance of working is provided by the following heuristic argument. If <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> has doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" />, then the Cartesian product <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> has doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^2}" class="latex" />. On the other hand, by using the projection map <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+%7B%5Cbf+F%7D_2%5En+%5Ctimes+%7B%5Cbf+F%7D_2%5En+%5Crightarrow+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+%7B%5Cbf+F%7D_2%5En+%5Ctimes+%7B%5Cbf+F%7D_2%5En+%5Crightarrow+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+%7B%5Cbf+F%7D_2%5En+%5Ctimes+%7B%5Cbf+F%7D_2%5En+%5Crightarrow+%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi: {\bf F}_2^n \times {\bf F}_2^n \rightarrow {\bf F}_2^n}" class="latex" /> defined by <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28x%2Cy%29+%3A%3D+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28x%2Cy%29+%3A%3D+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%28x%2Cy%29+%3A%3D+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi(x,y) := x+y}" class="latex" />, we see that <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> projects to <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28A+%5Ctimes+A%29+%3D+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%28A+%5Ctimes+A%29+%3D+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%28A+%5Ctimes+A%29+%3D+A%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi(A \times A) = A+A}" class="latex" />, with fibres <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E%7B-1%7D%28%5C%7Bh%5C%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E%7B-1%7D%28%5C%7Bh%5C%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%5E%7B-1%7D%28%5C%7Bh%5C%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi^{-1}(\{h\})}" class="latex" /> being essentially a copy of <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" />. So, morally, <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> also behaves like a “skew product” of <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> and the fibres <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" />, which suggests (non-rigorously) that the doubling constant <img src="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^2}" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ctimes+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \times A}" class="latex" /> is also something like the doubling constant of <img src="https://s0.wp.com/latex.php?latex=%7BA+%2B+A%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%2B+A%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%2B+A%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A + A}" class="latex" />, times the doubling constant of a typical fibre <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" />. This would imply that at least one of <img src="https://s0.wp.com/latex.php?latex=%7BA+%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A +A}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA+%5Ccap+%28A%2Bh%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A \cap (A+h)}" class="latex" /> would have doubling constant at most <img src="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K}" class="latex" />, and thus that at least one of operations (i), (ii) would not worsen the doubling constant.

Unfortunately, this argument does not seem to be easily made rigorous using the traditional doubling constant; even the significantly weaker statement that <img src="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%2BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A+A}" class="latex" /> has doubling constant at most <img src="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BK%5E2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{K^2}" class="latex" /> is false (see comments for more discussion). However, it turns out (as discussed in <a href="https://terrytao.wordpress.com/2023/06/27/sumsets-and-entropy-revisited/">this recent paper of myself with Green and Manners</a>) that things are much better. Here, the analogue of a subset <img src="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BA%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{A}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" /> is a random variable <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> taking values in <img src="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+F%7D_2%5En%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{{\bf F}_2^n}" class="latex" />, and the analogue of the (logarithmic) doubling constant <img src="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%7C%7D%7B%7CA%7C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%7C%7D%7B%7CA%7C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clog+%5Cfrac%7B%7CA%2BA%7C%7D%7B%7CA%7C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\log \frac{|A+A|}{|A|}}" class="latex" /> is the entropic doubling constant <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3A%3D+%7B%5Cbf+H%7D%28X_1%2BX_2%29-%7B%5Cbf+H%7D%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3A%3D+%7B%5Cbf+H%7D%28X_1%2BX_2%29-%7B%5Cbf+H%7D%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3A%3D+%7B%5Cbf+H%7D%28X_1%2BX_2%29-%7B%5Cbf+H%7D%28X%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X;X) := {\bf H}(X_1+X_2)-{\bf H}(X)}" class="latex" />, where <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,X_2}" class="latex" /> are independent copies of <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" />. If <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> is a random variable in some additive group <img src="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BG%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{G}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+G+%5Crightarrow+H%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+G+%5Crightarrow+H%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%3A+G+%5Crightarrow+H%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi: G \rightarrow H}" class="latex" /> is a homomorphism, one then has what we call the fibring inequality <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%3BX%29+%5Cgeq+d%28%5Cpi%28X%29%3B%5Cpi%28X%29%29+%2B+d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%2C&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%3BX%29+%5Cgeq+d%28%5Cpi%28X%29%3B%5Cpi%28X%29%29+%2B+d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%2C&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%3BX%29+%5Cgeq+d%28%5Cpi%28X%29%3B%5Cpi%28X%29%29+%2B+d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%2C&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle d(X;X) \geq d(\pi(X);\pi(X)) + d(X|\pi(X); X|\pi(X))," class="latex" />
where the conditional doubling constant <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X|\pi(X); X|\pi(X))}" class="latex" /> is defined as <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29+%3D+%7B%5Cbf+H%7D%28X_1+%2B+X_2+%7C+%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29+-+%7B%5Cbf+H%7D%28+X+%7C+%5Cpi%28X%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29+%3D+%7B%5Cbf+H%7D%28X_1+%2B+X_2+%7C+%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29+-+%7B%5Cbf+H%7D%28+X+%7C+%5Cpi%28X%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++d%28X%7C%5Cpi%28X%29%3B+X%7C%5Cpi%28X%29%29+%3D+%7B%5Cbf+H%7D%28X_1+%2B+X_2+%7C+%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29+-+%7B%5Cbf+H%7D%28+X+%7C+%5Cpi%28X%29+%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle d(X|\pi(X); X|\pi(X)) = {\bf H}(X_1 + X_2 | \pi(X_1), \pi(X_2)) - {\bf H}( X | \pi(X) )." class="latex" />
Indeed, from the chain rule for Shannon entropy one has <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X%29%29+%2B+%7B%5Cbf+H%7D%28X%7C%5Cpi%28X%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X%29%29+%2B+%7B%5Cbf+H%7D%28X%7C%5Cpi%28X%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X%29%29+%2B+%7B%5Cbf+H%7D%28X%7C%5Cpi%28X%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}(X) = {\bf H}(\pi(X)) + {\bf H}(X|\pi(X))" class="latex" />
and <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1%2BX_2%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%2B+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1%2BX_2%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%2B+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1%2BX_2%29+%3D+%7B%5Cbf+H%7D%28%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%2B+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}(X_1+X_2) = {\bf H}(\pi(X_1) + \pi(X_2)) + {\bf H}(X_1 + X_2|\pi(X_1) + \pi(X_2))" class="latex" />
while from the non-negativity of conditional mutual information one has <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%5Cgeq+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%5Cgeq+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29+%2B+%5Cpi%28X_2%29%29+%5Cgeq+%7B%5Cbf+H%7D%28X_1+%2B+X_2%7C%5Cpi%28X_1%29%2C+%5Cpi%28X_2%29%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle {\bf H}(X_1 + X_2|\pi(X_1) + \pi(X_2)) \geq {\bf H}(X_1 + X_2|\pi(X_1), \pi(X_2))" class="latex" />
and it is an easy matter to combine all these identities and inequalities to obtain the claim.

Applying this inequality with <img src="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X}" class="latex" /> replaced by two independent copies <img src="https://s0.wp.com/latex.php?latex=%7B%28X_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_1%2CX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_1,X_2)}" class="latex" /> of itself, and using the addition map <img src="https://s0.wp.com/latex.php?latex=%7B%28x%2Cy%29+%5Cmapsto+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28x%2Cy%29+%5Cmapsto+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28x%2Cy%29+%5Cmapsto+x%2By%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(x,y) \mapsto x+y}" class="latex" /> for <img src="https://s0.wp.com/latex.php?latex=%7B%5Cpi%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Cpi%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Cpi%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\pi}" class="latex" />, we obtain in particular that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%2CX_2%7CX_1%2BX_2%3B+X_1%2CX_2%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%2CX_2%7CX_1%2BX_2%3B+X_1%2CX_2%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%2CX_2%7CX_1%2BX_2%3B+X_1%2CX_2%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 2 d(X;X) \geq d(X_1+X_2; X_1+X_2) + d(X_1,X_2|X_1+X_2; X_1,X_2|X_1+X_2)" class="latex" />
or (since <img src="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_2}" class="latex" /> is determined by <img src="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1}" class="latex" /> once one fixes <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2}" class="latex" />) <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%5Cgeq+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 2 d(X;X) \geq d(X_1+X_2; X_1+X_2) + d(X_1|X_1+X_2; X_1|X_1+X_2)." class="latex" />
So if <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3D+%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3D+%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%3BX%29+%3D+%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X;X) = \log K}" class="latex" />, then at least one of <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%2BX_2%3B+X_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%2BX_2%3B+X_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X_1%2BX_2%3B+X_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X_1+X_2; X_1+X_2)}" class="latex" /> or <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X_1|X_1+X_2; X_1|X_1+X_2)}" class="latex" /> will be less than or equal to <img src="https://s0.wp.com/latex.php?latex=%7B%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%5Clog+K%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{\log K}" class="latex" />. This is the entropy analogue of at least one of (i) or (ii) improving, or at least not degrading the doubling constant, although there are some minor technicalities involving how one deals with the conditioning to <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2}" class="latex" /> in the second term <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X_1|X_1+X_2; X_1|X_1+X_2)}" class="latex" /> that we will gloss over here (one can pigeonhole the instances of <img src="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1}" class="latex" /> to different events <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2=x}" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%27%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%27%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%3Dx%27%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2=x'}" class="latex" />, and “depolarise” the induction hypothesis to deal with distances <img src="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Bd%28X%3BY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Bd%28X%3BY%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{d(X;Y)}" class="latex" /> between pairs of random variables <img src="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX%2CY%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X,Y}" class="latex" /> that do not necessarily have the same distribution). Furthermore, we can even calculate the defect in the above inequality: a careful inspection of the above argument eventually reveals that <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%3D+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%3D+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++2+d%28X%3BX%29+%3D+d%28X_1%2BX_2%3B+X_1%2BX_2%29+%2B+d%28X_1%7CX_1%2BX_2%3B+X_1%7CX_1%2BX_2%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle 2 d(X;X) = d(X_1+X_2; X_1+X_2) + d(X_1|X_1+X_2; X_1|X_1+X_2)" class="latex" />
<img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%7B%5Cbf+I%7D%28+X_1+%2B+X_2+%3A+X_1+%2B+X_3+%7C+X_1+%2B+X_2+%2B+X_3+%2B+X_4%29&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%7B%5Cbf+I%7D%28+X_1+%2B+X_2+%3A+X_1+%2B+X_3+%7C+X_1+%2B+X_2+%2B+X_3+%2B+X_4%29&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%2B+%7B%5Cbf+I%7D%28+X_1+%2B+X_2+%3A+X_1+%2B+X_3+%7C+X_1+%2B+X_2+%2B+X_3+%2B+X_4%29&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle + {\bf I}( X_1 + X_2 : X_1 + X_3 | X_1 + X_2 + X_3 + X_4)" class="latex" />
where we now take four independent copies <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%2CX_3%2CX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%2CX_3%2CX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2CX_2%2CX_3%2CX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1,X_2,X_3,X_4}" class="latex" />. This leads (modulo some technicalities) to the following interesting conclusion: if neither (i) nor (ii) leads to an improvement in the entropic doubling constant, then <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%7BX_2%2BX_3%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_2%2BX_3%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_2%2BX_3%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_2+X_3}" class="latex" /> are conditionally independent relative to <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2+X_3+X_4}" class="latex" />. This situation (or an approximation to this situation) is what we refer to in the paper as the “endgame”.

A version of this endgame conclusion is in fact valid in any characteristic. But in characteristic <img src="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B2%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{2}" class="latex" />, we can take advantage of the identity <img src="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X_1%2BX_2%29+%2B+%28X_2%2BX_3%29+%3D+X_1+%2B+X_3.&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X_1%2BX_2%29+%2B+%28X_2%2BX_3%29+%3D+X_1+%2B+X_3.&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%28X_1%2BX_2%29+%2B+%28X_2%2BX_3%29+%3D+X_1+%2B+X_3.&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="\displaystyle (X_1+X_2) + (X_2+X_3) = X_1 + X_3." class="latex" />
Conditioning on <img src="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7BX_1%2BX_2%2BX_3%2BX_4%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{X_1+X_2+X_3+X_4}" class="latex" />, and using symmetry we now conclude that if we are in the endgame exactly (so that the mutual information is zero), then the independent sum of two copies of <img src="https://s0.wp.com/latex.php?latex=%7B%28X_1%2BX_2%7CX_1%2BX_2%2BX_3%2BX_4%29%7D&bg=ffffff&fg=000000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%28X_1%2BX_2%7CX_1%2BX_2%2BX_3%2BX_4%29%7D&bg=ffffff&fg=000000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%28X_1%2BX_2%7CX_1%2BX_2%2BX_3%2BX_4%29%7D&bg=ffffff&fg=000000&s=0&c=20201002&zoom=4.5 4x" alt="{(X_1+X_2|X_1+X_2+X_3+X_4)}" class="latex" /> has exactly the same distribution; in particular, the entropic doubling constant here is zero, which is certainly a reduction in the doubling constant.

To deal with the situation where the conditional mutual information is small but not completely zero, we have to use an entropic version of the Balog-Szemeredi-Gowers lemma, but fortunately this was already worked out in an <a href="https://zbmath.org/1239.11015">old paper of mine</a> (although in order to optimise the final constant, we ended up using a slight variant of that lemma).

I am planning to formalize this paper in the <a href="https://leanprover-community.github.io/index.html">Lean4 language</a>. Further discussion of this project will take place on <a href="https://leanprover.zulipchat.com/#narrow/stream/412902-Polynomial-Freiman-Ruzsa-conjecture">this Zulip stream</a>, and the project itself will be held at <a href="https://github.com/teorth/pfr">this Github repository.</a>

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