High density piecewise syndeticity of product sets in amenable groups

TL;DR

This paper establishes a quantitative version of a known result, showing that product sets in amenable groups with positive density are piecewise syndetic, and provides explicit lower bounds on the density of witnesses to this property.

Contribution

It introduces a quantitative bound on the density of witnesses for the piecewise syndeticity of product sets in amenable groups, extending previous qualitative results.

Findings

Provides a lower bound on the density of witnesses in amenable groups

Extends results to higher-dimensional groups like bZ^d

Quantifies the thickness of product sets in amenable groups

Abstract

M. Beiglb\"ock, V. Bergelson, and A. Fish proved that if $G$ is a countable amenable group and $A$ and $B$ are subsets of $G$ with positive Banach density, then the product set $AB$ is piecewise syndetic. This means that there is a finite subset $E$ of $G$ such that $EAB$ is thick, that is, $EAB$ contains translates of any finite subset of $G$. When $G=\mathbb{Z}$, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a F\o lner sequence) of the set of witnesses to the thickness of $% EAB$. When $G=\mathbb{Z}^d$, this result was first proven by the current set of authors using completely different techniques.