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 $\mathrm{\mid }A\mathrm{\mid }$. We prove that the minimal period of such a tiling is bounded by $\exp (c(\log D{)}^2\mathrm{/}\log \log D)$, where $D$ is the diameter of $A$. In the converse direction, given $\epsilon >0$, we construct tilings whose minimal period has the same prime factors as $\mathrm{\mid }A\mathrm{\mid }$ and is bounded from below by $D^{3\mathrm{/}2-\epsilon }$. We also discuss the relationship between minimal tiling period estimates and the Coven-Meyerowitz conjecture.