Riesz bases, Meyer's quasicrystals, and bounded remainder sets

Abstract

We consider systems of exponentials with frequencies belonging to simple quasicrystals in ${\mathbb{R}}^d$. We ask if there exist domains $S$ in ${\mathbb{R}}^d$ which admit such a system as a Riesz basis for the space $L^2(S)$. We prove that the answer depends on an arithmetical condition on the quasicrystal. The proof is based on the connection of the problem to the discrepancy of multi-dimensional irrational rotations, and specifically, to the theory of bounded remainder sets. In particular it is shown that any bounded remainder set admits a Riesz basis of exponentials. This extends to several dimensions (and to the non-periodic setting) the results obtained earlier in dimension one.

To appear in Transactions of the AMS