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Approximate invariance for ergodic actions of amenable groups. (English) Zbl 1432.37003
Summary: 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 $$(\mathbb{Z},+)$$, 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.

##### MSC:
 37A05 Dynamical aspects of measure-preserving transformations 37A25 Ergodicity, mixing, rates of mixing 37A30 Ergodic theorems, spectral theory, Markov operators 22D40 Ergodic theory on groups
##### Keywords:
action sets; aperiodicity; density theorems
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