Plunnecke inequalities in countable abelian groups

Plunnecke inequalities for sumsets of finite sets in abelian groups are extended to measure-preserving systems (mps). For a set A in a group, and a set B of positive measure in mps, we estimate the measure of the union of translations along the set A of B. As an application, through Furstenberg's correspondence principle, we obtain the new Plunnecke type inequalities for lower and upper Banach density in countable abelian groups. Based on joint works with Michael Bjorklund, Chalmers, and Kamil Bulinski, Sydney.

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