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Progression-free sets in $$\mathbb{Z}_4^n$$ are exponentially small. (English) Zbl 1425.11019
This paper represents without a doubt one of the most important breakthroughs in discrete mathematics in 2016. This is despite (or possibly to some extent because of) its relative brevity and elementary nature.
Given a finite abelian group $$G$$, let $$r_k(G)$$ denote the cardinality of a largest subset of $$G$$ containing no non-trivial $$k$$-term arithmetic progression. The case when $$k=3$$ and $$G$$ is an $$n$$-dimensional vector space over a finite field of small fixed characteristic $$p$$ is considered a toy problem for Roth’s problem in the integers [B. Green, in: Surveys in combinatorics 2005, Lond. Math. Soc. Lect. Note Ser. 327, 1–27 (2005; Zbl 1155.11306)], but it is also of independent interest to the design theory and theoretical computer science communities, amongst others.
R. Meshulam [J. Comb. Theory, Ser. A 71, No. 1, 168–172 (1995; Zbl 0832.11006)] proved that $r_3(\mathbb F^n_3) \ll 3^n/n,$ using a Fourier iteration argument going back to [K. F. Roth, J. Lond. Math. Soc. 28, 104–109 (1953; Zbl 0050.04002)] which is now considered standard in the field. Nudging this upper bound toward the lower bound of $3^{0.72485n} \approx 2.2174^n$ provided by a construction of Y. Edel [Des. Codes Cryptography 31, No. 1, 5–14 (2004; Zbl 1057.51005)] has proved surprisingly difficult. It was only in 2012 that M. D. Bateman and N. H. Katz [J. Am. Math. Soc. 25, No. 2, 585–613 (2012; Zbl 1262.11010)], in a paper that is by many considered a technical tour de force, managed to improve the exponent of the denominator from $$1$$ to $$1+\varepsilon$$ for some small but explicit $$\varepsilon >0$$.
Finite abelian groups $$G$$ of even order were first considered by V. F. Lev J. Number Theory 104, No. 1, 162–169 (2004; Zbl 1043.11022)], who adapted Meshulam’s proof to show that $r_3(G) < 2|G|/\mathrm{rank}(2G).$ T. Sanders [Anal. PDE 2, No. 2, 211–234 (2009; Zbl 1197.11017)] improved upon this for the specific group $$G=(\mathbb Z/4\mathbb Z)^n$$, showing that $r_3((\mathbb Z/4\mathbb Z)^n) \ll 4n/(n\log^\varepsilon n)$ for some absolute constant $$\varepsilon >0$$, using a more sophisticated (but still Fourier-analytic) iteration technique.
The main result of the present paper is the following.
Theorem. Let $$n\ge 1$$ and suppose that $$A\subseteq(\mathbb Z/4\mathbb Z)^n$$ contains no non-trivial arithmetic progression of length $$3$$. Then $|A| \le 4^{\gamma n}$ for a constant $$0<\gamma<1$$.
Specifically, the constant $$\gamma\approx 0.926$$ is obtained as the maximum over $$0<\varepsilon <1/4$$ of $$\tfrac12(H(0.5-\varepsilon)+H(2\varepsilon))$$, where $$H$$ is the binary entropy function.
This result is of significance for at least two reasons: first, it represents an exponential improvement over previous work, bringing the upper bound for the first time within reasonable reach of the lower bound; second, it spawned a flurry of further results in the immediate aftermath of its publication. Arguably the most important of these to date is the paper by J. S. Ellenberg and D. C. Gijswijt [Ann. Math. (2) 185, No. 1, 339–343 (2017; Zbl 1425.11020)], which reduces the upper bound on $$r_3(\mathbb F^n_3)$$ to roughly $$2.756^n$$, alongside a handful of others that had not been formally published at the time that this review was written.
The core contribution of Croot, Lev and Pach is the realisation that a version of the polynomial method can be used to tackle Roth-type problems in certain finite abelian groups. For a comprehensive survey on the polynomial method, its variants and their applications, see [T. C. Tao, EMS Surv. Math. Sci. 1, No. 1, 1–46 (2014; Zbl 1294.05044)]. One recent application that stands out – having surfaced as unexpectedly as the result of the present paper – is the resolution of the finite-field Kakeya conjecture by Z. Dvir [J. Am. Math. Soc. 22, No. 4, 1093–1097 (2009; Zbl 1202.52021)].
The key ingredient in the proof of the above theorem is the following simple linear-algebraic lemma, also used in the aforementioned subsequent work of Ellenberg and Gijswijt: If $$P$$ is a multilinear polynomial in $$n$$ variables of total degree at most $$d$$ over a field $$F$$ such that $$P(a-b)=0$$ for all $$a\ne b\in A$$, then $$P(-a)$$ cannot be non-zero for too many elements $$a\in A$$.
This lemma can be used to prove that if $$A$$ is a progression-free subset of $$\mathbb Z/4\mathbb Z)^n$$ then there are few $$F_n$$-cosets containing many elements of $$A$$, where $$F_n$$ denotes the subgroup of $$\mathbb Z/4\mathbb Z)^n$$ generated by its involutions (which is isomorphic to $$(\mathbb Z/4\mathbb Z)^n))$$. From this the bound on the size of $$A$$ follows essentially by averaging and the tensor-power trick.
Reviewer: Julia Wolf

##### MSC:
 11B25 Arithmetic progressions 11B13 Additive bases, including sumsets
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##### References:
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