On the minimal period of integer tilings

Abstract If a finite set tiles the integers by translations, it also admits a tiling whose period has the same prime factors as . We prove that the minimal period of such a tiling is bounded by , where is the diameter of . In the converse direction, given , we construct tilings whose minimal period has the same prime factors as and is bounded from below by . We also discuss the relationship between minimal tiling period estimates and the Coven–Meyerowitz conjecture.