The ultraproduct construction is a useful tool in model theory to study the asymptotic behavior of a class of structures. In the particular case of a class of finite groups, the ultralimit of the normalized counting measure yields a translation-invariant Keisler measure on internal sets, which has played a crucial role in the recent years in several applications of model-theoretic techniques to additive combinatorics.

In this talk, we present a model-theoretic result that resonates with Croot-Sisask's almost periodicity technique for a general group equipped with a Keisler measure under some mild assumptions. We then show how to use this result to obtain, via an ultrafilter construction, a non-quantitative proof of Roth’s theorem on arithmetic progressions of length three. The core idea of our model-theoretic version of almost periodicity is the stability-like behaviour of a convolution of sets. We will not assume prior knowledge of model theory for this talk.

In the first part of the talk, aimed at a general (non-logic) audience, we will recall the ultraproduct construction of finite groups, as well as Łoś's theorem, dense internal subsets and the main features of stable relations, in order to briefly outline how to prove a non-quantitative version of Roth's theorem.

The second part of the talk will focus on a more detailed explanation of some aspects of the proofs, in particular the notions of dense and random elements and their features. If time permits, we will explain how some of these techniques can be adapted to study the collection of starting points of arithmetic progressions in the primes and in the square-free integers.