On exponential frames near the critical density

TL;DR

This paper constructs exponential frames with near-critical density for $L^2$ spaces on sets in $ $, providing bounds close to the optimal and extending results to locally compact abelian groups.

Contribution

It establishes the existence of exponential frames with density arbitrarily close to the critical density, with explicit frame bounds, solving a longstanding problem.

Findings

Existence of exponential frames with density $D( abla) o | ext{Omega}|$

Frame bounds depend only on $ ext{epsilon}$, not on the set

Extension of results to locally compact abelian groups

Abstract

Given a relatively compact set $\Omega \subseteq \mathbb{R}$ of Lebesgue measure $|\Omega|$ and $\varepsilon > 0$, we show the existence of a set $\Lambda \subseteq \mathbb{R}$ of uniform density $D (\Lambda) \leq (1+\varepsilon) |\Omega|$ such that the exponential system $\{ \exp(2\pi i \lambda \cdot) \mathbf{1}_{\Omega}: \lambda \in \Lambda \}$ is a frame for $L^2 (\Omega)$ with frame bounds $A |\Omega|, B |\Omega|$ for constants $A,B$ only depending on $\varepsilon$. This solves a problem on the frame bounds of an exponential frame near the critical density posed by Nitzan, Olevskii and Ulanovskii. We also prove an extension to locally compact abelian groups, which improves a result by Agora, Antezana and Cabrelli by providing frame bounds involving the spectrum.