Given an abelian group \(G\), a corner is a subset of pairs of the form \((x,y),(x+d,y),(x,y+d)\) with \(d\neq 0\) non trivial. Ajtai and Szemerédi proved that, asymptotically, every dense subset \(S\) of \({\mathbb Z}/N{\mathbb Z} \times {\mathbb Z}/N{\mathbb Z}\) contains a corner. Shkredov gave a quantitative lower bound on the density of the subset \(S\) for particular finite abelian groups. In this talk, we will explain how model-theoretic conditions on the subset \(S\), such as stability, imply the existence of corners and of other combinatorial configurations for (pseudo)-finite abelian groups.
Corners, Squares and Stability
10.10.2024 15:00 - 15:50
Organiser:
KGRC
Location: