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{\bf Mihail N. Kolountzakis}
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{\bf Translational Tilings of the Integers with Long Periods}
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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.
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