Discrete Uniqueness Sets for Functions with Spectral Gaps

It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\mathrm{\Lambda }$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having “periodic gaps”. The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets $\mathrm{\Lambda }$ satisfy a strong non-uniqueness property: Every discrete function $c(\lambda )\in l^2(\mathrm{\Lambda })$ can be interpolated by an analytic $L^2$-function with spectrum in $S$.