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 Λ satisfy a strong non-uniqueness property: every discrete function c(λ)\in l^2(Λ) can be interpolated by an analytic L^2-function with spectrum in S.Bibliography: 9 titles.