  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 $S$.
