A probabilistic technique for finding almost-periods of convolutions

TL;DR

This paper introduces a probabilistic method for identifying almost-periods in convolutions of group subsets, enabling new results in additive combinatorics without relying on Fourier analysis.

Contribution

It presents a novel probabilistic technique for almost-periods, extending additive combinatorics results to non-abelian groups and sparse sets without Fourier transforms.

Findings

Probabilistic proof of Roth's theorem on 3-term arithmetic progressions

Variant of Bourgain-Green theorem for sparse sumsets

Structural results for product sets in non-abelian groups

Abstract

We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, ..., N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A B C and A^2 A^{-2} are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in product sets A B. Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.