Exponential frames and Riesz sequences at the critical density

TL;DR

The paper solves the problem of characterizing exponential frames and Riesz sequences at the critical density on sets of finite measure by linking them to weak limits of translates and hulls. It establishes that a frame (resp. a Riesz sequence) at the critical density occurs if and only if some weak limit of translates yields a Riesz basis, thus connecting density conditions to hull-based limit objects. The authors also derive a counterexample showing that no critical-density frame or Riesz sequence exists for a specific Kozma–Nitzan–Olevskii set, and they prove that for repetitive or almost repetitive point sets the critical-density frames/Riesz sequences are already Riesz bases. These results generalize Seip’s interval-based theory to general finite-measure sets and settle open questions posed by Olevskii, with implications for cut-and-project and other aperiodic structures.

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.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of \ref{['thm:noframe_critical']}
• Theorem 1.3
• proof
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 3.1
• Lemma 3.2