Multi-tiling and Riesz bases

Let S   be a bounded, Riemann measurable set in RdRd, and Λ be a lattice. By a theorem of Fuglede, if S   tiles RdRd with translation set Λ, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles  RdRd with translation set Λ, S has a Riesz basis of exponentials. The proof is based on Meyerʼs quasicrystals.