Multiple lattice tiles and Riesz bases of exponentials

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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 $\Lambda$.