Gabriel Conant


University of Cambridge


Monday, April 12, 2021 - 4:00pm to 5:30pm



In 2011, Malliaris and Shelah proved that finite stable graphs satisfy a strengthened version of Szemeredi’s Regularity Lemma, with polynomial bounds and no irregular pairs. In 2017, Terry and Wolf proved an analogue of this for stable subsets of finite abelian groups, based on Green’s “arithmetic regularity lemma". Roughly speaking, they showed that stable subset of a finite abelian group can be approximated by a union of cosets of a subgroup whose index is bounded by a exponential function depending only on the stability constant and approximation error. These results for abelian groups were qualtitatively generalized to all finite groups by C., Pillay, and Terry, and then to finite subsets of arbitrary groups by Martin-Pizarro, Palacin, and Wolf. However, the generalizations for non-abelian groups used model-theoretic techniques involving ultraproducts, and thus produced no explicit quantitative bounds. In this talk, I will discuss a new proof of these results, which avoids the use of ultraproducts and yields effective bounds. These techniques also improve the bound in the abelian case from exponential to polynomial, and yield the Polynomial Freiman-Ruzsa Conjecture for finite stable subsets of arbitrary groups.