Olof Sisask: Almost-periodicity of convolutions: methods from, and applications to, additive combinatorics

Abstract: If A is a large subset of {1,...,N}, then how long an arithmetic progression must A+A = {a+b : a,b in A} contain? And why would one be interested in this combinatorial question from an analytic perspective? It turns out that analysis is a very natural tool in this context, and some very nice answers were given by Bourgain and then Green, both using Fourier-analytic techniques. In this talk I shall describe how one can tackle the problem using the almost-periodicity of convolutions, as well as how one can obtain such almost-periodicity results using ideas from discrete geometry and probability.
Joint work with Ernie Croot and Izabella Laba.