On exponential frames near the critical density

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.