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arXiv · 2406.14824

On the minimal period of integer tilings

Izabella Łaba·Dmitrii Zakharov
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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 $ε>0$, we construct tilings whose minimal period has the same prime factors as $|A|$ and is bounded from below by $D^{3/2-ε}$. We also discuss the relationship between minimal tiling period estimates and the Coven-Meyerowitz conjecture.

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Izabella Łaba, Dmitrii Zakharov. 2024-07-03. On the minimal period of integer tilings. https://arxiv.org/abs/2406.14824

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