It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $Lambda$. 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 $Lambda$ 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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