Quasi-periodic tiling with multiplicity: a lattice enumeration approach

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 $\Lambda$ 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 that a convex polytope $P$ that 1-tiles $\mathbb R^d$ with a discrete multiset $\Lambda$ 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, Robins, and Shiryaev, which states that if $P$ $k$-tiles $\mathbb R^d$ with a discrete multiset $\Lambda$, 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 $\Lambda$ such that every element of $\Lambda$ 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, Kolountzakis, Robins, and Shiryaev, showing that for $d \in \{2,3\}$, if a polytope $P$ $k$-tiles $\mathbb R^d$ with a discrete multiset $\Lambda$, 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 $\Lambda$ 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.