Exponential frames and Riesz sequences at the critical density

TL;DR

This paper characterizes when exponential systems at critical density form frames or Riesz sequences, linking these properties to the existence of weak limits that produce Riesz bases, and resolves an open problem in the field.

Contribution

It provides a complete characterization of exponential frames and Riesz sequences at critical density using weak limits, and demonstrates the existence of sets lacking these properties, solving an open problem.

Findings

Existence of sets without exponential frames or Riesz sequences at critical density.

Exponential frames and Riesz sequences over repetitive sets at critical density are Riesz bases.

Characterization of systems via weak limits that yield Riesz bases.

Abstract

We characterize exponential systems on sets of finite measure that form a frame or a Riesz sequence at the critical density. Namely, they are precisely those systems for which the underlying point set admits a weak limit that yields a Riesz basis. In combination with a recent result by Kozma, Nitzan and Olevskii, this shows that there exist sets that fail to possess a frame or Riesz sequence at the critical density, solving an open problem posed by Olevskii. As another consequence, we show that exponential frames and Riesz sequences over repetitive point sets (such as cut-and-project sets) at the critical density are already Riesz bases.