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

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 |

### Citations:

Zbl 0051.28104
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\textit{M. Björklund} et al., J. Éc. Polytech., Math. 8, 311--329 (2021; Zbl 1467.37002)

### References:

[1] | Björklund, Michael; Fish, Alexander, Approximate invariance for ergodic actions of amenable groups, Discrete Anal., 56 p. pp. (2019) · Zbl 1432.37003 |

[2] | Boshernitzan, Michael; Wierdl, Máté, Ergodic theorems along sequences and Hardy fields, Proc. Nat. Acad. Sci. U.S.A., 93, 16, 8205-8207 (1996) · Zbl 0863.28011 |

[3] | Bourgain, J., On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math., 61, 1, 39-72 (1988) · Zbl 0642.28010 |

[4] | Kneser, Martin, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z., 58, 459-484 (1953) · Zbl 0051.28104 |

[5] | Kneser, Martin, Summenmengen in lokalkompakten abelschen Gruppen, Math. Z., 66, 88-110 (1956) · Zbl 0073.01702 |

[6] | Lacroix, Y., Natural extensions and mixing for semi-group actions, Publ. Inst. Rech. Math. Rennes, 2, 10 p. pp. (1995) |

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