Product set phenomena for countable groups

We develop in this paper novel 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. Our methods heavily utilize the theory of C*-algebras and random walks on groups. As applications of our methods, we extend and quantify a series of recent results by Jin, Bergelson–Furstenberg–Weiss, Beiglböck–Bergelson–Fish, Griesmer and Di Nasso–Lupini to general countable groups.