Higher-order Generalisations of Stability and Arithmetic Regularity

Joint work with Caroline Terry initiated during a Simons programme in 2017 identified model-theoretic stability as a sufficient condition for the existence of strong arithmetic regularity decompositions in finite abelian groups, as pioneered by Ben Green around 2003. Higher-order arithmetic regularity decompositions, based on Tim Gowers’s groundbreaking work on Szemerédi’s theorem in the late 90s, are an essential part of today';s arithmetic combinatorics toolkit and have found multiple applications in theoretical computer science.

In this talk, I will describe recent joint work with Caroline Terry in which we define a natural higher-order generalisation of stability and prove that it implies the existence of particularly efficient higher-order arithmetic regularity decompositions in the setting of finite elementary abelian p-groups.

NOTE: This talk was presented live and a recording will not be available.