Exponential Riesz bases, discrepancy of irrational rotations and BMO

TL;DR

This paper investigates when exponential systems with frequencies from quasicrystals form Riesz bases in L^2, linking diophantine conditions to basis properties and extending discrepancy results to BMO.

Contribution

It establishes a diophantine condition as necessary and sufficient for Riesz basis property and extends Kesten's discrepancy theorem to BMO.

Findings

Diophantine condition characterizes Riesz basis property

Extension of Kesten's theorem to BMO

Connection between quasicrystals and basis properties

Abstract

We study the basis property of systems of exponentials with frequencies belonging to 'simple quasicrystals'. We show that a diophantine condition is necessary and sufficient for such a system to be a Riesz basis in L^2 on a finite union of intervals. For the proof we extend to BMO a theorem of Kesten about the discrepancy of irrational rotations of the circle.