Sets of k-recurrence but not (k+1)-recurrence
[Ensembles de k-récurrence mais pas de k+1-récurrence]
Annales de l'Institut Fourier, Tome 56 (2006) no. 4, pp. 839-849.

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, 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.

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
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)
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     title = {Sets of $k$-recurrence but not $(k+1)$-recurrence},
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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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