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

2016/02/01 by Grepstad, Sigrid, Lev, Nir · 2 citations
#11K38 #42C15 #52C23 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1602.00495

Abstract

We consider systems of exponentials with frequencies belonging to simple quasicrystals in ℝ^d. We ask if there exist domains S in ℝ^d which admit such a system as a Riesz basis for the space L²(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.

Cited by

• Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability
• Exponential frames and Riesz sequences at the critical density

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