# Definable sets containing productsets in expansions of groups

## TL;DR

This paper investigates conditions under which definable sets in expansions of groups contain productsets, linking model-theoretic stability, density, and regularity lemmas to characterize largeness and productset properties.

## Contribution

It establishes the productset property for definable sets with positive Banach density in stable expansions of amenable groups and introduces a $1$-sided version for arbitrary expansions, connecting it to model-theoretic notions.

## Key findings

- Productset property holds for stable, positive Banach density sets in discrete amenable groups.
- A $1$-sided productset property is characterized using coheir independence.
- Regularity lemmas in distal theories enable definable productset properties in certain expansions.

## Abstract

We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the "productset property"). We first show that the productset property holds for any definable subset $A$ of an expansion of a discrete amenable group such that $A$ has positive Banach density and the formula $x\cdot y\in A$ is stable. For arbitrary expansions of groups, we consider a "$1$-sided" version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this $1$-sided version, and behaves as a notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity. Finally, we use recent work on regularity lemmas in distal theories to prove a definable version of the productset property for sets of positive Banach density definable in certain distal expansions of amenable groups.