Translational Tilings of the Integers with Long Periods

Abstract

Suppose that $A \subseteq {\Bbb{Z}}$ is a finite set of integers of diameter $D=\max A - \min A$. Suppose also that $B \subseteq {\Bbb{Z}}$ is such that $A\oplus B = {\Bbb{Z}}$, that is each $n\in{\Bbb{Z}}$ is uniquely expressible as $a+b$, $a\in A$, $b\in B$. We say then that $A$ tiles the integers if translated at the locations $B$ and it is well known that $B$ must be a periodic set in this case and that the smallest period of $B$ is at most $2^D$. Here we study the relationship between the diameter of $A$ and the least period ${\cal P}(B)$ of $B$. We show that ${\cal P}(B) \le c_2 \exp(c_3 \sqrt D \log D \sqrt{\log\log D})$ and that we can have ${\cal P}(B) \ge c_1 D^2$, where $c_1, c_2, c_3 > 0$ are constants.