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 $\left|A\right|$. We prove that the minimal period of such a tiling is bounded by $\exp (c{\left(\log D\right)}^2/\log \log D)$, 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 $\left|A\right|$ and is bounded from below by $D^{3/2-\epsilon }$. We also discuss the relationship between minimal tiling period estimates and the Coven–Meyerowitz conjecture.