Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle T. A set of integers is called r-Bohr if it is recurrent for all products of r rotations on T, and Bohr if it is recurrent for all products of rotations on T. It is a result due to Katznelson that for each r 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 Z, and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.