Multiple lattice tiles and Riesz bases of exponentials

Abstract

Suppose $Ω\subseteq\RR^d$ is a bounded and measurable set and $Λ\subseteq \RR^d$ is a lattice. Suppose also that $Ω$ tiles multiply, at level $k$, when translated at the locations $Λ$. This means that the $Λ$-translates of $Ω$ cover almost every point of $\RR^d$ exactly $k$ times. We show here that there is a set of exponentials $\exp(2πi t\cdot x)$, $t\in T$, where $T$ is some countable subset of $\RR^d$, which forms a Riesz basis of $L^2(Ω)$. This result was recently proved by Grepstad and Lev under the extra assumption that $Ω$ 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 $Λ^*$. It can be chosen knowing only $Λ$ and $k$ and is the same for all $Ω$ that tile multiply with $Λ$.