Progression-free sets in $\mathbb{Z}_4^n$ are exponentially small

Abstract

Abstract. We show that for integer n ≥ 1, any subset A ⊆ Z n 4 free of three-term arithmetic progressions has size |A| ≤ 4 γn , with an absolute constant γ ≈ 0.926. Background and MotivationIn his influential papers [R52, R53], Roth has shown that if a set A ⊆ {1, 2, . . . , N } does not contain three elements in an arithmetic progression, then |A| = o(N ) and indeed, |A| = O(N/ log log N ) as N grows. Since then, estimating the largest possible size of such a set has become one of the central problems in additive combinatorics. 4 / log N ). It is easily seen that Roth's problem is essentially equivalent to estimating the largest possible size of a subset of the cyclic group Z N , free of three-term arithmetic progressions. This makes it natural to investigate other finite abelian groups.We say that a subset A of an (additively written) abelian group G is progression-free if there do not exist pairwise distinct a, b, c ∈ A with a + b = 2c, and we denote by r 3 (G) the largest size of a progression-free subset A ⊆ G. For abelian groups G of odd order, Brown and Buhler [BB82] and independently Frankl, Graham, and Rödl [FGR87] proved that r 3 (G) = o(|G|) as |G| grows. Meshulam [M95], following the general lines of Roth's argument, has shown that if G is an abelian group of odd order, then r 3 (G) ≤ 2|G|/ rk(G) (where we use the standard notation rk(G) for the rank of G); in particular, r 3 (Z n m ) ≤ 2m n /n. Despite many efforts, no further progress was made for over 15 years, till Bateman and Katz in their ground-breaking paper [BK12] proved that r 3 (Z n 3 ) = O(3 n /n 1+ε ) with an absolute constant ε > 0.Abelian groups of even order were first considered in [L04] where, as a further elaboration on the Roth-Meshulam proof, it is shown that r 3 (G) < 2|G|/ rk(2G) for any finite abelian group G; here 2G = {2g : g ∈ G}. For the homocyclic groups of exponent 4 this †

BME VIK Annual Research Report on Electrical Engineering and Computer Science 2016

“…For the homocyclic groups of exponent 4 this result was improved by Sanders, who proved that r 3 (Z 4 n ) = O(4 n /n(log n) ε ) with an absolute constant ε>0. Croot, Lev and Pach [169] further improved Sanders's result. They showed that r 3 (Z 4 n ) = O(3.62 n ).…”

“…Starting from Roth, the standard way to obtain quantitative estimates for r 3 (G) involves a combination of the Fourier analysis and the density increment technique. In contrast, in [169] a new variant of the polynomial method is developed and used without resorting to the familiar Fourier analysis-density increment strategy.…”

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“…As we mentioned at the end of Section 3.1, by recent breakthrough work [23,27], such subsets A ⊂ F n 3 have polynomially small density, |A| < (3 n ) 1−ε for some positive constant ε.…”

“…Subsequently, Bateman and Katz [7] improved the bound to |A| ≤ (c/n 1+ε ) · 3 n for some small unspecified constant ε > 0. Very recently, breakthrough works of Croot, Lev and Pach [23] and Ellenberg and Gijswijt [27] improved the bound to c n for c < 3 an absolute constant. Their techniques are quite different; we refer the interested reader to the papers for details.…”

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“…First, Croot, Lev, and Pach obtained an upper bound of this exponential type for subsets of Z n 4 with no 3-term arithmetic progression [12]. Then, barely a week later, Jordan Ellenberg and Dion Gijswijt independently saw how to modify the argument of Croot, Lev, and Pach to give a similar bound for the cap-set problem itself, thereby giving a positive answer to the question above [13].…”