Universal completeness of exponentials

We consider generalizations of the classical Fourier uniqueness theorem. First, we construct a family of uniformly discrete sets $\mathrm{\Lambda }\subset \mathbb{R}$, of uniform density one, such that the exponential system $\{e^{2\pi i\lambda x}:\lambda \in \mathrm{\Lambda }\}$ is complete in $L^p(S)$ for every $1\le p<\mathrm{\infty }$ and every measurable set $S\subset \mathbb{R}$ with $\mathrm{\mid }S\mathrm{\mid }<1$. We also show that no set that is asymptotically integer can have this universality property. Additionally, for every $v\in (0,1)$, we construct a set of integer frequencies and uniform density $v$ whose exponential system is complete in $L^p(S)$ for every $1\le p<\mathrm{\infty }$ and every measurable set $S\subset [0,1]$ with $\mathrm{\mid }S\mathrm{\mid }<v$. Finally, we prove that the Sobolev regularity condition $\alpha >\frac{1}{2}$ for the existence of uniformly discrete uniqueness sets for spectra with periodic weak gaps, considered by Olevskii and Ulanovskii, is sharp. Our findings admit extensions to higher dimensions and are verified in Lean.