Corners and stability

by Prof. Amador Martin-Pizarro (Albert-Ludwigs-Universität Freiburg)

Thursday 14 Dec 2023, 11:00 → 12:00 Europe/Rome

Aula Riunioni (Dipartimento di Matematica)

Given an abelian group $G$, a corner is a a subset of pairs of the form $(x,y),(x+g,y),(x,y+g)$ with $g$ non trivial. Ajtai and Szemerédi proved that, asymptotically for finite abelian groups, every dense subset $S$ of $G×G$ contains an corner. Shkredov gave a quantitative lower bound on the density of the subset $S$. In this talk, we will explain how model-theoretic conditions on the subset $S$, such as local stability, will imply the existence of corners and of other configurations for (pseudo-)finite abelian groups. This is joint work with D. Palacin (Madrid) and J. Wolf (Cambridge).