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

TL;DR

This paper investigates when systems of exponential functions with frequencies from quasicrystals form Riesz bases in $L^2$ spaces, linking the problem to bounded remainder sets and multidimensional discrepancy theory, and extending previous one-dimensional results.

Contribution

It establishes a connection between Riesz bases of exponentials and bounded remainder sets in multiple dimensions, providing new conditions based on arithmetical properties of quasicrystals.

Findings

Bounded remainder sets admit Riesz bases of exponentials.

The existence of Riesz bases depends on arithmetical conditions of the quasicrystal.

Extension of one-dimensional results to higher dimensions and non-periodic settings.

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.