Uniqueness sets for unbounded spectra

For every set $S?\mathbb{R}$ of finite measure, we construct a system of exponentials ${\{e^{i\lambda t}\}}_{\lambda \in \Lambda }$ which is complete in $L^2(S)$ and such that the set of frequencies Λ has the critical density $D(\Lambda )=\mathrm{mes}(S)/2\pi$.