Geodesic
Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 603-627
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We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $\mathbb T$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $\mathbb T$, and Bohr if it is recurrent for all products of rotations on $\mathbb T$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on $\mathbb Z$, and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.
We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $\mathbb T$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $\mathbb T$, and Bohr if it is recurrent for all products of rotations on $\mathbb T$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on $\mathbb Z$, and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.
DOI : 10.1007/s10587-013-0043-z
Classification : 37A45, 37B05, 37B20
Keywords: recurrence for dynamical systems; non-recurrence for dynamical systems; rotations of the unit circle; syndetic set; Bohr topology on $\mathbb {Z}$; Bohr set; $r$-Bohr set 
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Grivaux, Sophie; Roginskaya, Maria. Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 3, pp. 603-627. doi: 10.1007/s10587-013-0043-z

[1] Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211 (2007), 766-793. | DOI | MR | Zbl

[2] Bergelson, V., Junco, A. Del, Lemańczyk, M., Rosenblatt, J.: Rigidity and non-recurrence along sequences. Preprint 2011, arXiv:1103.0905.

[3] Bergelson, V., Haland, I. J.: Sets of recurrence and generalized polynomials. V. Bergelson, et al. Convergence in Ergodic Theory and Probability Papers from the conference, Ohio State University, Columbus, OH, USA, June 23-26, 1993, de Gruyter, Berlin, Ohio State Univ. Math. Res. Inst. Publ. 5 (1996), 91-110. | MR | Zbl

[4] Bergelson, V., Lesigne, E.: Van der Corput sets in $\mathbb Z^d$. Colloq. Math. 110 (2008), 1-49. | DOI | MR

[5] Boshernitzan, M., Glasner, E.: On two recurrence problems. Fundam. Math. 206 (2009), 113-130. | DOI | MR | Zbl

[6] Furstenberg, H.: Poincaré recurrence and number theory. Bull. Am. Math. Soc., New Ser. 5 (1981), 211-234. | DOI | MR | Zbl

[7] Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, N. J. (1981). | MR | Zbl

[8] Glasner, E.: On minimal actions of Polish groups. Topology Appl. 85 (1998), 119-125. | DOI | MR | Zbl

[9] Glasner, E.: Classifying dynamical systems by their recurrence properties. Topol. Methods Nonlinear Anal. 24 (2004), 21-40. | DOI | MR | Zbl

[10] Glasner, E., Weiss, B.: On the interplay between measurable and topological dynamics. Handbook of Dynamical Systems vol. 1B B. Hasselblatt et al. Amsterdam, Elsevier (2006), 597-648. | MR | Zbl

[11] Grivaux, S.: Non-recurrence sets for weakly mixing dynamical systems. (to appear) in Ergodic Theory Dyn. Syst., | DOI

[12] Kannan, R., Lovász, L.: Covering minima and lattice-point-free convex bodies. Ann. Math. (2) 128 (1988), 577-602. | MR | Zbl

[13] Katznelson, Y.: Chromatic numbers of Cayley graphs on $\mathbb Z$ and recurrence. Combinatorica 21 (2001), 211-219. | DOI | MR

[14] Kříž, I.: Large independent sets in shift-invariant graphs. Graphs Comb. 3 (1987), 145-158. | DOI | MR | Zbl

[15] Pestov, V.: Forty-plus annotated questions about large topological groups. Open Problems in Topology II Elliott M. Pearl Elsevier, Amsterdam (2007), 439-450. | MR

[16] Petersen, K.: Ergodic Theory. Cambridge Studies in Advanced Mathematics 2 Cambridge University Press, Cambridge (1983). | MR | Zbl

[17] Queffélec, M.: Substitution Dynamical Systems. Spectral analysis. 2nd ed. Lecture Notes in Mathematics 1294. Springer, Berlin (2010). | MR | Zbl

[18] Veech, W. A.: The equicontinuous structure relation for minimal abelian transformation groups. Am. J. Math. 90 (1968), 723-732. | DOI | MR | Zbl

[19] Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics vol. 79 Springer, New-York (1982). | DOI | MR | Zbl

[20] Weiss, B.: Single Orbit Dynamics. CBMS Regional Conference Series in Math. 95. AMS, Providence (2000). | MR | Zbl

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