Sets of k-recurrence but not (k+1)-recurrence
[Ensembles de k-récurrence mais pas de k+1-récurrence]
Frantzikinakis, Nikos ; Lesigne, Emmanuel ; Wierdl, Máté
Annales de l'Institut Fourier, Tome 56 (2006), p. 839-849 / Harvested from Numdam
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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∈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.

Publié le : 2006-01-01
DOI : https://doi.org/10.5802/aif.2202
Classification:  38A,  11B
Mots clés: théorie ergodique, récurrence, récurrence multiple, combinatoire additive des nombres 
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     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},
     volume = {56},
     year = {2006},
     pages = {839-849},
     doi = {10.5802/aif.2202},
     zbl = {1123.37001},
     mrnumber = {2266880},
     language = {en},
     url = {http://dml.mathdoc.fr/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) pp. 839-849. doi : 10.5802/aif.2202. http://gdmltest.u-ga.fr/item/AIF_2006__56_4_839_0/

[1] Bergelson, V. Weakly mixing PET, Ergodic Theory Dynamical Systems, Tome 7 (1987) no. 3, pp. 337-349 | MR 912373 | Zbl 0645.28012

[2] Bergelson, V. Ergodic Ramsey theory-an update, Ergodic Theroy of ℤ d -actions, Cambridge University Press, Cambridge (1996), pp. 1-61 | MR 1411215 | Zbl 0846.05095

[3] Furstenberg, H. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., Tome 71 (1977), pp. 204-256 | Article | MR 498471 | Zbl 0347.28016

[4] Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J. (1981) (M. B. Porter Lectures) | MR 603625 | Zbl 0459.28023

[5] Furstenberg, H.; Katznelson, Y. An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math., Tome 34 (1979), pp. 275-291 | Article | MR 531279 | Zbl 0426.28014

[6] Furstenberg, H.; Katznelson, Y.; Ornstein, D. The ergodic theoretical proof of Szemerédi’s theorem, Bull. Amer. Math. Soc. (N.S.), Tome 7 (1982) no. 3, pp. 527-552 | Article | MR 670131 | Zbl 0523.28017

[7] Host, B.; Kra, B. Nonconventional ergodic averages and nilmanifolds, Annals of Math, Tome 161 (2005) no. 1, pp. 397-488 | Article | MR 2150389 | Zbl 1077.37002

[8] Katznelson, Y. Chromatic numbers of Cayley graphs on ℤ and recurrence, Combinatorica, Tome 21 (2001) no. 2, pp. 211-219 (Paul Erdős and his mathematics (Budapest, 1999)) | Article | MR 1832446 | Zbl 0981.05038

[9] Ziegler, Tamar Universal Characteristic Factors and Furstenberg Averages, http://www.arxiv.org/abs/math.DS/0403212 (To appear in Journal of the AMS)