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. We show that the same is true for a much wider range of 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 uniformly discrete sets $\Lambda$ satisfy a strong non-uniqueness property: every discrete function $c(\lambda)\in l^2(\Lambda)$ can be interpolated by an analytic $L^2$-function with spectrum in $S$.
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