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

Grepstad, Sigrid; Lev, Nir

doi:10.1090/tran/7157

Riesz bases, Meyer’s quasicrystals, and bounded remainder sets
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Trans. Amer. Math. Soc. 370 (2018), 4273-4298 Request permission

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.

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