Complete type amalgamation for non-standard finite groups

TL;DR

This paper generalizes key results in model theory and additive combinatorics, providing new measure-theoretic and model-theoretic proofs for properties of finite groups and arithmetic progressions.

Contribution

It extends Hrushovski's stabilizer theorem, generalizes Gowers' results, and offers new model-theoretic proofs for classical combinatorial theorems in finite and amenable groups.

Findings

Proves a measure-theoretic version of Pillay-Scanlon-Wagner's result.

Provides model-theoretic proofs of Gowers' and Roth's theorems.

Establishes asymptotic results for quasirandom and amenable groups.

Abstract

We extend previous work on Hrushovski's stabilizer's theorem and prove a measure-theoretic version of a well-known result of Pillay-Scanlon-Wagner on products of three types. This generalizes results of Gowers on products of three sets and yields model-theoretic proofs of existing asymptotic results for quasirandom groups. We also obtain a model-theoretic proof of Roth's theorem on the existence of arithmetic progressions of length $3$ for subsets of positive density in suitable definably amenable groups, such as countable amenable abelian groups without involutions and ultraproducts of finite abelian groups of odd order.