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Periodic structure of translational multi-tilings in the plane
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 143, Number 6, December 2021
- pp. 1841-1862
- 10.1353/ajm.2021.0047
- Article
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abstract:
Suppose $f\in L^1(\Bbb{R}^d)$, $\Lambda\subset\Bbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with a weight. We prove that there exists a lattice $L\subset\Bbb{R}^d$ such that $f+L$ also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity.
Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficient criteria to determine whether a convex polygon admits translational multi-tilings. As an application, one can easily construct symmetric $(2m)$-gons, for any $m\geq 4$, that do not multi-tile by translations.
Finally, we prove a convex polygon which is not a parallelogram only admits periodic multiple tilings, if any.