PERIODIC STRUCTURE OF TRANSLATIONAL MULTI-TILINGS IN THE PLANE

Bochen Liu

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Suppose f ∈ L1(Rd), Λ ⊂ Rd is a finite union of translated lattices such that f +Λ tiles with a weight. We prove that there exists a lattice L ⊂ Rd 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≥ 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.

Original languageEnglish
Pages (from-to)1841-1862
Number of pages22
JournalAmerican Journal of Mathematics
Volume143
Issue number6
DOIs
StatePublished - Dec 2021
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2021 by Johns Hopkins University Press.

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