zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv