Plünnecke inequalities for countable abelian groups

Michael Bjorklund, Alexander Fish

Abstract

We establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of countable abelian groups. 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, via a correspondence principle for product sets, Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group. In the special case of the integers and bases with respect to the upper Banach density, this extends recent results by R. Jin.

Keywords: Ergodic Ramsey theory, additive combinatorics.

AMS Subject Classification: Primary 37B05; secondary 11B13, 11K70.