  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.
