On Roth's theorem on progressions

Abstract

We show that if A is a subset of {1,...,N} contains no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{1-o(1)} N). The approach is somewhat different from that used in arXiv:1007.5444.Comment: 16 pp. Corrected the proof of the Croot-Sisask Lemma. Corrected typos. Updated reference

Nearly complete graphs decomposable into large induced matchings and their applications

“…. , N } with no 3-term arithmetic progressions-a problem that received a considerable amount of attention over the years (see [25] and its references).…”

A More Efficient Computationally Sound Non-Interactive Zero-Knowledge Shuffle Argument

“…On the other hand, it is known that r 3 (N ) = O(N (log log N ) 5 / log N ) [San11]. Thus, according to [San11], the minimal N such that r 3 (N ) = n is ω(n), while according to Elkin, N = O(n2 2 √ 2 log 2 n ) = n 1+o(1) .…”

“…Thus, according to [San11], the minimal N such that r 3 (N ) = n is ω(n), while according to Elkin, N = O(n2 2 √ 2 log 2 n ) = n 1+o(1) . Thus, for any fixed n > 0, there exists N = n 1+o(1) , such that [N ] contains an n-element progression-free subset [Lip12].…”

Ramsey multiplicity of linear patterns in certain finite abelian groups

“…Giving a lower bound for t 3−AP (A) in terms of the density of A corresponds to the infamously difficult problem of obtaining upper bounds in Roth's theorem (see for example Chapter 10 of [34]). In particular, it is known through recent work of Bloom [1] (building on prior work of Sanders [28]…”