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.

Reçu le : 2005-04-01
Accepté le : 2005-07-25
DOI : https://doi.org/10.5802/aif.2202
Classification : 38A,  11B
Mots clés: théorie ergodique, récurrence, récurrence multiple, combinatoire additive des nombres
@article{AIF_2006__56_4_839_0,
     author = {Frantzikinakis, Nikos and Lesigne, Emmanuel and Wierdl, M\'at\'e},
     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'institut Fourier},
     volume = {56},
     number = {4},
     year = {2006},
     doi = {10.5802/aif.2202},
     zbl = {1123.37001},
     mrnumber = {2266880},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2006__56_4_839_0/}
}
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. https://aif.centre-mersenne.org/item/AIF_2006__56_4_839_0/

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