A Plünnecke–Ruzsa inequality in compact abelian groups

Abstract

The Plünnecke–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 Plünnecke–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.