Approximate subgroups

TL;DR

This paper explores the structure of approximate subgroups within definably amenable groups, establishing the existence of a special type-definable subgroup with measure-theoretic properties.

Contribution

It introduces a new structural result linking approximate subgroups and type-definable subgroups in first-order structures.

Findings

Existence of a type-definable subgroup normalized by the approximate subgroup

The subgroup is contained within the fourth power of the approximate subgroup

Every definable superset of this subgroup has positive measure

Abstract

Given a definably amenable approximate subgroup $A$ of a (local) group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ such that every definable superset of $H$ has positive measure.