Quasi-periodic Tiling with Multiplicity: A Lattice Enumeration Approach

Abstract

The k-tiling problem for a convex polytope P is the problem of covering $\mathbb {R}^d$ with translates of P using a discrete multiset $\varLambda$ of translation vectors such that every point in $\mathbb {R}^d$ is covered exactly k times, except possibly for the boundary of P and its translates. A classical result in the study of tiling problems is a theorem of McMullen [Mathematika 27(1):113–121, 1980] that a convex polytope P that 1-tiles $\mathbb {R}^d$ with a discrete multiset $\varLambda$ can, in fact, 1-tile $\mathbb {R}^d$ with a lattice $\mathcal {L}$. A generalization of McMullen’s theorem for k-tiling was conjectured by Gravin et al. [Combinatorica 32(6):629–649, 2012], which states that if P k-tiles $\mathbb {R}^d$ with a discrete multiset $\varLambda$, then P m-tiles $\mathbb {R}^d$ with a lattice $\mathcal {L}$ for some m. In this paper, we consider the case when P k-tiles $\mathbb {R}^d$ with a discrete multiset $\varLambda$ such that every element of $\varLambda$ is contained in a quasi-periodic set $\mathcal {Q}$ (i.e., a finite union of translated lattices). This is motivated by the result of Gravin et al. [Discrete Comput Geom 50(4):1033–1050, 2013] and Kolountzakis [Discrete Comput Geom 23(4):537–553, 2000], showing that for $d \in \{2,3\}$, if a polytope P k-tiles $\mathbb {R}^d$ with a discrete multiset $\varLambda$, then P m-tiles $\mathbb {R}^d$ with a quasi-periodic set $\mathcal {Q}$ for some m. Here we show for all values of d that if a polytope P k-tiles $\mathbb {R}^d$ with a discrete multiset $\varLambda$ that is contained in a quasi-periodic set $\mathcal {Q}$ that satisfies a mild hypothesis, then P m-tiles $\mathbb {R}^d$ with a lattice $\mathcal {L}$ for some m. This strengthens the results of Gravin, Kolountzakis, Robins, and Shiryaev, and is a step in the direction of proving the conjecture of Gravin et al. [Combinatorica 32(6):629–649, 2012].