The structure of stable sets: from additive number theory to model theory

A long-standing open problem in additive number theory is the following: how large does a subset of the cyclic group modulo a (very large) prime have to be before it is guaranteed to contain a non-trivial arithmetic progression of length 3? 

 

In the first half of this talk we shall survey recent progress on this problem, and the techniques used to solve it and related questions about additive structures in finite abelian groups. In particular, we shall explain the idea behind the so-called "arithmetic regularity lemma" pioneered by Green, which is a group-theoretic analogue of Szemerédi's celebrated regularity lemma for graphs.

 

In the second half of the talk we shall describe recent joint work with Caroline Terry (University of Maryland), which shows that under the natural model-theoretic assumption of stability the conclusions of the arithmetic regularity lemma can be significantly strengthened, leading to a characterisation of stable subsets of finite-dimensional vector spaces over finite fields.

 

This talk will not assume any particular background knowledge, and should be accessible to postgraduate students across pure mathematics.