Pl\"unnecke inequalities for countable abelian groups

Abstract

We establish in this paper a new form of Pl\"unnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers. We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Pl\"unnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.