Sets of transfer times with small densities

Michael Björklund; Alexander Fish; Ilya D. Shkredov

doi:10.5802/jep.147

Sets of transfer times with small densities
[Ensembles de temps de transfert avec petites densités]

Michael Björklund ¹ ; Alexander Fish ² ; Ilya D. Shkredov ³

¹ Department of Mathematics, Chalmers University Chalmers Tvärgata 3, SE-412 96 Gothenburg, Sweden
² The School of Mathematics and Statistics, Faculty of Science, The University of Sydney NSW 2006, Sydney, Australia
³ Steklov Mathematical Institute ul. Gubkina 8, 119991 Moscow, Russia

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 311-329.

Résumé
Abstract

Dans cet article, nous introduisons et discutons plusieurs notions de doublement pour des actions préservant la mesure sur un groupe abélien dénombrable $G$ . Notre résultat principal caractérise les actions $2$ -doublantes et peut être vu comme une extension de nature ergodique de certains théorèmes de densité classiques pour les sommes d’ensembles par Kneser. Tous nos résultats sont optimaux et sont nouveaux déjà pour le cas où $G = (ℤ, +)$ .

In this paper we introduce and discuss various notions of doubling for measure-preserving actions of a countable abelian group $G$ . Our main result characterizes $2$ -doubling actions, and can be viewed as an ergodic-theoretical extension of some classical density theorems for sumsets by Kneser. All of our results are completely sharp and they are new already in the case when $G = (ℤ, +)$ .

Reçu le : 2020-05-04
Accepté le : 2021-01-26
Publié le : 2021-02-10

MR Zbl

DOI : 10.5802/jep.147

Classification : 37A44, 28D05, 11B13
Keywords: Return times, inverse theorems, sumsets
Mots-clés : Temps de retour, théorèmes inverses, sommes d’ensembles

Affiliations des auteurs :

Michael Björklund ¹ ; Alexander Fish ² ; Ilya D. Shkredov ³

¹ Department of Mathematics, Chalmers University Chalmers Tvärgata 3, SE-412 96 Gothenburg, Sweden
² The School of Mathematics and Statistics, Faculty of Science, The University of Sydney NSW 2006, Sydney, Australia
³ Steklov Mathematical Institute ul. Gubkina 8, 119991 Moscow, Russia

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Sets of transfer times with small densities},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {311--329},
     publisher = {\'Ecole polytechnique},
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Michael Björklund; Alexander Fish; Ilya D. Shkredov. Sets of transfer times with small densities. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 311-329. doi : 10.5802/jep.147. https://jep.centre-mersenne.org/articles/10.5802/jep.147/

Bibliographie
Cité par

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[4] M. Kneser - “Abschätzung der asymptotischen Dichte von Summenmengen”, Math. Z. 58 (1953), p. 459-484 | DOI | Zbl

[5] M. Kneser - “Summenmengen in lokalkompakten abelschen Gruppen”, Math. Z. 66 (1956), p. 88-110 | DOI | MR | Zbl

[6] Y. Lacroix - “Natural extensions and mixing for semi-group actions”, Publ. Inst. Rech. Math. Rennes 2 (1995), article ID 7, 10 pages | MR

A. Fish, L. Jiang I. D. Shkredov - “Direct and inverse results for popular differences in trees of positive dimension”, Ergodic Theory and Dynamical Systems 44 (2024) no. 2, p. 481-508 | DOI:10.1017/etds.2023.18 | Zbl:1535.37006

Cité par 1 document. Sources : zbMATH