The Solecki submeasures and densities on groups

TL;DR

This paper introduces the Solecki submeasure and density on groups, exploring their relationship with Haar measure and extending classical results on difference and sumsets to all amenable groups.

Contribution

It defines the Solecki submeasure and density, analyzes their properties, and generalizes key results from classical harmonic analysis and additive combinatorics to amenable groups.

Findings

Solecki submeasure and density are introduced and studied.

Right Solecki density coincides with upper Banach density on countable amenable groups.

Classical results on difference sets and sumsets are extended to all amenable groups.

Abstract

We introduce the Solecki submeasure $\sigma(A)=\inf_F\sup_{x,y\in G}|F\cap xAy|/|F|$ and its left and right modifications on a group $G$, and study the interplay between the Solecki submeasure and the Haar measure on compact topological groups. Also we show that the right Solecki density on a countable amenable group coincides with the upper Banach density $d^*$ which allows us to generalize some fundamental results of Bogoliuboff, Folner, Cotlar and Ricabarra, Ellis and Keynes about difference sets and Jin, Beiglbock, Bergelson and Fish about the sumsets to the class of all amenable groups.