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On the structure of multiple translational tilings by polygonal regions

Mihail N. Kolountzakis
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Abstract

We consider polygons with the following ``pairing property'': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon $K$ tiles the plane multiply when translated at the locations $Λ$, where $Λ$ is a multiset in the plane. The pairing property of $K$ makes this question particularly amenable to Fourier Analysis. After establishing a necessary and sufficient condition for $K$ to tile with a given lattice $Λ$ (which was first found by Bolle for the case of convex polygons-notice that all convex polygons that tile, necessarily have the pairing property and, therefore, our theorems apply to them) we move on to prove that a large class of such polygons tiles only quasi-periodically, which for us means that $Λ$ must be a finite union of translated 2-dimensional lattices in the plane. For the particular case of convex polygons we show that all convex polygons which are not parallelograms tile necessarily quasi-periodically, if at all.

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Mihail N. Kolountzakis. 1999-04-14. On the structure of multiple translational tilings by polygonal regions. https://arxiv.org/abs/math/9904065

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