Uncertainty Principle, annihilating pairs and Fourier restriction

Let $G$ be a locally compact abelian group, and let $\widehat{G}$ denote its dual group, equipped with a Haar measure. A variant of the uncertainty principle states that for any $S\subset G$ and $\mathrm{\Sigma }\subset \widehat{G}$, there exists a constant $C(S,\mathrm{\Sigma })$ such that for any $f\in L^2(G)$, the following inequality holds: $$\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(G)}\le C(S,\mathrm{\Sigma })(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(G\setminus S)}+\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{L^2(\widehat{G}\setminus \mathrm{\Sigma })}),$$ where $\widehat{f}$ denotes the Fourier transform of $f$. This variant of the uncertainty principle is particularly useful in applications such as signal processing and control theory.The purpose of this paper is to show that such estimates can be strengthened when $S$ or $\mathrm{\Sigma }$ satisfies a restriction theorem and to provide an estimate for the constant $C(S,\mathrm{\Sigma })$. This result serves as a quantitative counterpart to a recent finding by the first and last author. In the setting of finite groups, the results also extend those of Matolcsi-Szücs and Donoho-Stark.