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

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, Volume 56 (2006) no. 4, pp. 839-849.

Abstract
Résumé

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.

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.

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

Author's affiliations:

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, Volume 56 (2006) no. 4, pp. 839-849. doi : 10.5802/aif.2202. http://archive.numdam.org/articles/10.5802/aif.2202/

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