# Article

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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
Summary:
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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