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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
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