Approximate invariance for ergodic actions of amenable groups
  We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups.   As an application of these techniques, we prove a dynamical generalization of Kneser's celebrated density theorem for subsets in $(\bZ,+)$, valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study. 
Michael Bj\"orklund and Alexander Fish
math.DS math.GR math.NT