Product set phenomena for countable groups

Michael Bjorklund, Alexander Fish

Abstract

We develop in this paper general techniques to analyze local combinatorial structures in product sets of two subsets of a countable group which are "large" with respect to certain classes of (not necessarily invariant) means on the group. As applications of our methods, we extend and quantify a series of recent results by Jin, Bergelson-Furstenberg-Weiss, Beiglböck-Bergelson-Fish, Griesmer and diNasso-Lupini to general countable groups.

AMS Subject Classification: Primary: 37B05; Secondary: 05C81, 11B13, 11K70 .