Growth in infinite groups of infinite subsets

TL;DR

This paper explores measure-theoretic properties of infinite groups, demonstrating that certain combinatorial results from finite groups extend to infinite groups with finitely additive measures, revealing new structural insights.

Contribution

It introduces a finitely additive measure on infinite groups that parallels finite group subset sizes, extending combinatorial results like Ruzsa distance to infinite settings.

Findings

Infinite groups with infinitely many finite index subgroups have subsets with measure close to 1/2.

Such subsets can have their product set measure less than 1.

The measure shares properties with subset sizes in finite groups.

Abstract

Given an infinite group G, we consider the finitely additive measure defined on finite unions of cosets of finite index subgroups. We show that this shares many properties with the size of subsets of a finite group, for instance we can obtain equivalent results on Ruzsa distance and product free sets. In particular if G has infinitely many finite index subgroups then it has subsets S of measure arbitrarily close to 1/2 with the square of S having measure less than 1.