Progression-free sets in Z₄ⁿ are exponentially small

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 Motivation Inhis 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. Roth's original results were improved by Heath-Brown H87, Szemerédi S90, Bourgain B99, Sanders S12, S11, and Bloom B, the current record due to Bloom being |A| = O (N (log log N) 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 n3) = 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