Multiple lattice tiles and Riesz bases of exponentials

TL;DR

This paper proves that for multiply tiling sets with a lattice, there exists a Riesz basis of exponential functions, using an elementary linear algebra approach, generalizing previous results that required boundary measure conditions.

Contribution

It establishes the existence of Riesz bases of exponentials for multiply tiling sets using a simpler, more elementary method than prior quasicrystal-based proofs.

Findings

The set of frequencies T is a finite union of shifted dual lattice copies.

T can be explicitly constructed from the lattice and tiling level.

The same T works for all sets tiling multiply with the same lattice.

Abstract

Suppose $\Omega\subseteq\RR^d$ is a bounded and measurable set and $\Lambda \subseteq \RR^d$ is a lattice. Suppose also that $\Omega$ tiles multiply, at level $k$, when translated at the locations $\Lambda$. This means that the $\Lambda$-translates of $\Omega$ cover almost every point of $\RR^d$ exactly $k$ times. We show here that there is a set of exponentials $\exp(2\pi i t\cdot x)$, $t\in T$, where $T$ is some countable subset of $\RR^d$, which forms a Riesz basis of $L^2(\Omega)$. This result was recently proved by Grepstad and Lev under the extra assumption that $\Omega$ has boundary of measure 0, using methods from the theory of quasicrystals. Our approach is rather more elementary and is based almost entirely on linear algebra. The set of frequencies $T$ turns out to be a finite union of shifted copies of the dual lattice $\Lambda^*$. It can be chosen knowing only $\Lambda$ and $k$ and is the same for all $\Omega$ that tile multiply with $\Lambda$.