A Pl\"unnecke-Ruzsa inequality in compact abelian groups

TL;DR

This paper extends the Pl"unnecke-Ruzsa inequality to measurable subsets of compact abelian groups using Haar measure, focusing on K-analytic sets and exploring implications in descriptive topology.

Contribution

It introduces a Pl"unnecke-Ruzsa inequality for K-analytic sets in compact abelian groups, broadening its applicability beyond finite groups.

Findings

Established inequality for K-analytic sets in compact abelian groups

Discussed stability of Haar measurable sets under addition

Explored extensions with implications in descriptive topology

Abstract

The Pl\"unnecke-Ruzsa inequality is a fundamental tool to control the growth of finite subsets of abelian groups under repeated addition and subtraction. Other tools to handle sumsets have gained applicability by being extended to more general subsets of more general groups. This motivates extending the Pl\"unnecke-Ruzsa inequality, in particular to measurable subsets of compact abelian groups by replacing the cardinality with the Haar probability measure. This objective is related to the question of the stability of classes of Haar measurable sets under addition. In this direction the class of analytic sets is a natural one to work with. We prove a Pl\"unnecke-Ruzsa inequality for K-analytic sets in general compact (Hausdorff) abelian groups. We also discuss further extensions, some of which raise questions of independent interest in descriptive topology.