On the minimal period of integer tilings

TL;DR

This paper investigates the minimal period of integer tilings, establishing bounds related to the set size and diameter, and explores connections to the Coven-Meyerowitz conjecture.

Contribution

It provides new bounds on the minimal period of integer tilings and constructs examples with large minimal periods, advancing understanding of tiling periodicity.

Findings

Minimal period bounded by exponential of squared log diameter

Existence of tilings with minimal period at least D^{3/2 - ε}

Discussion of links to the Coven-Meyerowitz conjecture

Abstract

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 $\exp(c(\log D)^2/\log\log D)$, 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 $|A|$ 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.