On the minimal period of integer tilings

If a finite set A tiles the integers by translations, it also admits a tiling whose period M has the same prime factors as $|A|$. We prove that the minimal period of such a tiling is bounded by exp(c(log D)²/loglog D), where D is the diameter of A. In the converse direction, given ε>0, we construct tilings whose minimal period has the same prime factors as $|A|$ and is bounded from below by D3/2-ε. We also discuss the relationship between minimal tiling period estimates and the Coven-Meyerowitz conjecture.