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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
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.
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