On exponential frames near the critical density
Marcin Bownik, Jordy Timo van Velthoven
TL;DR
The paper resolves the long-standing question of constructing exponential frames near the critical density with explicit, ε-dependent frame bounds. It blends a refined finite-dimensional sparsification approach (building on Batson–Spielman–Srivastava theory and scalable-frames) with lifting techniques to extend from the real line to general LCA groups, using elemental reductions and quasi-dyadic cubes. A central contribution is the explicit dependence of frame bounds on the spectrum, yielding A(ε) and B(ε) tied to μ̂(Ω), and a density guarantee D(Λ)≤(1+ε)|Ω|. The work also bridges Beurling/Leptin density theories to Landau-type conditions, removing compact-generation limitations and providing a robust framework for exponential frames on broad group settings with practical density-control implications.
Abstract
Given a relatively compact set $Ω\subseteq \mathbb{R}$ of Lebesgue measure $|Ω|$ and $\varepsilon > 0$, we show the existence of a set $Λ\subseteq \mathbb{R}$ of uniform density $D (Λ) \leq (1+\varepsilon) |Ω|$ such that the exponential system $\{ \exp(2πi λ\cdot) \mathbf{1}_Ω: λ\in Λ\}$ is a frame for $L^2 (Ω)$ with frame bounds $A |Ω|, B |Ω|$ 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.