Sets of $k$-recurrence but not $(k+1)$-recurrence

Frantzikinakis, Nikos; Lesigne, Emmanuel; Wierdl, Máté

doi:10.5802/aif.2202

Sets of

k

-recurrence but not

(k + 1)

-recurrence
[Ensembles de

k

-récurrence mais pas de

k + 1

-récurrence]

Frantzikinakis, Nikos ¹ ; Lesigne, Emmanuel ² ; Wierdl, Máté ³

¹ Pennsylvania State University Department of Mathematics McAllister Building University Park, PA 16802 (USA)
² Université François Rabelais de Tours Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083) Faculté des Sciences et Techniques Parc de Grandmont 37200 Tours (France)
³ University of Memphis Department of Mathematical Sciences Memphis, TN 38152 (USA)

Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 839-849.

Résumé
Abstract

Pour tout nombre entier $k > 0$ , nous construisons un ensemble d’entiers qui est un ensemble de récurrence multiple à l’ordre $k$ mais pas à l’ordre $k + 1$ . Cela étend une construction de Furstenberg qui a construit un ensemble de récurrence qui n’est pas un ensemble de 2-récurrence. Nous obtenons un résultat similaire pour la convergence des moyennes ergodiques multiples. Comme conséquence de notre construction, nous exhibons aussi un résultat combinatoire relié au théorème de Szemerédi.

For every $k \in ℕ$ , we produce a set of integers which is $k$ -recurrent but not $(k + 1)$ -recurrent. This extends a result of Furstenberg who produced a $1$ -recurrent set which is not $2$ -recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.

EuDML MR Zbl

DOI : 10.5802/aif.2202

Classification : 38A, 11B
Keywords: Ergodic theory, recurrence, multiple recurrence, combinatorial additive number theory
Mot clés : théorie ergodique, récurrence, récurrence multiple, combinatoire additive des nombres

Affiliations des auteurs :

Frantzikinakis, Nikos ¹ ; Lesigne, Emmanuel ² ; Wierdl, Máté ³

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     title = {Sets of $k$-recurrence but not $(k+1)$-recurrence},
     journal = {Annales de l'Institut Fourier},
     pages = {839--849},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Frantzikinakis, Nikos; Lesigne, Emmanuel; Wierdl, Máté. Sets of $k$-recurrence but not $(k+1)$-recurrence. Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 839-849. doi : 10.5802/aif.2202. http://www.numdam.org/articles/10.5802/aif.2202/

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