Olof Sisask: Finding long words in short products of approximate subgroups, via the almost-periodicity of convolutions

The finite subgroups of a group G can be characterised as the finite subsets A of G that contain the identity and do not grow under multiplication: \(|A.A| = |A|\). But what happens if we relax this growth condition, and only ask that the growth is limited, for example that \(|A.A| < 1000 |A|\)? What can we say about such `approximate subgroups'?

Here we shall focus on the problem of finding long words in A, or in related sets. In the case when A is a subgroup, we can take k arbitrary elements of A, multiply them, and the product will still lie in A. That is, A on its own contains arbitrarily long words from A. We shall show that even when A is an approximate subgroup, we can still find very long words in the short product \(A.A.A.A\), with letters coming from a reasonably large subset of A. This is in fact one of the ingredients in the deep classification of approximate subgroups by Breuillard, Green and Tao.

These results on statistically weakened subgroups are in fact intimately tied to results in analysis, and we shall see how a mix of analysis and probability can be used to tackle them. Indeed, we shall see how results like the above follow very naturally from results on the almost-periodicity of convolutions, and we shall see how a probabilistic argument can be used to prove such analytic results. This method can in fact substitute nicely even in the non-abelian setting for the use of the Fourier transform that would be a natural port of call for abelian groups. No familiarity with the area will be assumed!

Based in part on joint work with E. Croot.