Speaker: Gabriel Conant (University of Cambridge)
Title: Quantitative stable arithmetic regularity in arbitrary finite groups
Abstract: In 2011, Malliaris and Shelah showed that “stable” finite graphs admit a strong form of Szemeredi’s regularity lemma with polynomial bounds and no irregular pairs. Here “stable” is a combinatorial property, motivated by model theory, which is defined by omitting so-called “half-graphs”. In 2017, Terry and Wolf developed an analogue of this result for subsets of finite abelian groups, which is based on the notion of arithmetic regularity as invented by Green. Roughly speaking, they showed that any stable subset of a finite abelian group can be efficiently approximated by cosets of a subgroup whose index is bounded exponentially in the approximation error and the stability constant. Around the same time, in joint work with Pillay and Terry, we proved a version of this for arbitrary finite groups using model theoretic techniques, which resulted in stronger qualitative features, but lacked any information about quantitative bounds. In this talk, I will discuss a new effective proof of our result, which yields quantitative bounds for arbitrary finite groups, and improves the bound in Terry & Wolf’s result from exponential to polynomial.
