Near optimal bounds in Freiman's theorem

Abstract

We prove that if for a finite set A of integers we have |A + A| ≤ K|A|, then A is contained in a generalized arithmetic progression of dimension at most K 1+C(log K) −1/2 and size at most exp(K 1+C(log K) −1/2)|A| for some absolute constant C. We also discuss a number of applications of this result.

On Sumsets of Convex Sets

“…which has recently found many applications [8,10,11,12,13]. Roughly speaking, we show that if |A − A| is not much bigger than |A| 3/2 then the energy E(A, A − A) is very large.…”

New bounds on cap sets

“…Even Ruzsa's original version [R99] suffices. We also mention a remarkable recent result in the integer setting due to Schoen [Sch11]. with > 0 to be determined later.…”

The Erdős–Moser Sum-free Set Problem

“…Although it was more unusual at the time of [SSV05], it is now common-place to apply the Balog-Szemerédi-Gowers-Freiman machinery in this sort of situation. This tells us that if A Ă Z and EpAq ě η|A| 3 , then there is an arithmetic progression P such that [Sch11]. For our purposes arithmetic progressions are the same as intervals and the final ingredient we need is the following.…”