Sets of k-recurrence but not (k+1)-recurrence
Annales de l'Institut Fourier, Volume 56 (2006) no. 4, p. 839-849

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.

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.

DOI : https://doi.org/10.5802/aif.2202
Classification:  38A,  11B
Keywords: Ergodic theory, recurrence, multiple recurrence, combinatorial additive number theory
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {4},
     year = {2006},
     pages = {839-849},
     doi = {10.5802/aif.2202},
     mrnumber = {2266880},
     zbl = {1123.37001},
     language = {en},
     url = {http://www.numdam.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, Volume 56 (2006) no. 4, pp. 839-849. doi : 10.5802/aif.2202. http://www.numdam.org/item/AIF_2006__56_4_839_0/

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