On the Annihilation of Thin Sets

One says that a pair of sets $(S,Q)$ in $\mathbb{R}$ is 'annihilating' if no function can be concentrated on $S$ while having its Fourier transform concentrated on $Q$. One uses to distinguish between weak and strong annihilation types. It is well known that if both sets $S$ and $Q$ are of finite measure then they are strongly annihilating. In this paper we prove that if $S$ is a set of finite measure with periodic gaps, and $Q$ is a set of density zero, then weak annihilation holds. On the other hand a counter-example is constructed, showing that strong annihilation, in general, does not.