Sets of transfer times with small densities. (Ensembles de temps de transfert avec petites densités.) (English. French summary) Zbl 1467.37002

Björklund, Michael; Fish, Alexander; Shkredov, Ilya D.

J. Éc. Polytech., Math. 8, 311-329 (2021).

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.

Reviewer: Thomas B. Ward (Leeds)

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

Keywords:

return times; inverse theorems; sumsets

Citations:

Zbl 0051.28104

References:

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