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Sets of transfer times with small densities. (Ensembles de temps de transfert avec petites densités.) (English. French summary) Zbl 07315958
For an ergodic measure-preserving action of a discrete countable abelian group \(G\) on a probability space \((X,\mu)\) together with a sequence \((F_n)\) of finite subsets of \(G\) with the property that the pointwise ergodic theorem holds for averaging along \((F_n)\) (which gives a natural notion of lower asymptotic density \(\underline{d}\) for subsets of \(G\)), the authors define the set of transfer times \({\mathscr{R}}_{A,B}=\{g\in G\mid\mu(A\cap g^{-1}B)>0\}\) for measurable sets \(A,B\) with \(\mu(A)+\mu(B)<1\). The main results are aimed at establishing lower bounds for \(\underline{d}({\mathscr{R}}_{A,B})\) and to address questions about when such lower bounds are attained. The final results are sharp, and new even in the classical setting \(G=\mathbb{Z}\) in part because the hypothesis needed on the sequence \((F_n)\) is not that it is a Følner sequence, but that it admits the ergodic theorem. The arguments show that these questions are related to the “small doubling phenomenon” in additive combinatorics, and some of the results are seen as ergodic-theoretic extensions of M. Kneser’s theorem [Math. Z. 58, 459–484 (1953; Zbl 0051.28104)] concerning the lower asymptotic density of sumsets in the natural numbers.
MSC:
37A15 General groups of measure-preserving transformations and dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37A30 Ergodic theorems, spectral theory, Markov operators
37A44 Relations between ergodic theory and number theory
28D05 Measure-preserving transformations
11B13 Additive bases, including sumsets
22F05 General theory of group and pseudogroup actions
22F10 Measurable group actions
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