Uncertainty Principle, annihilating pairs and Fourier restriction
Philippe Jaming, Alexander Iosevich, Azita Mayeli
TL;DR
The paper develops a quantitative framework linking Fourier restriction estimates to an enhanced uncertainty principle via annihilating pairs on locally compact abelian groups. By requiring that the spectral set $ Sigma$ satisfies a $(p,q)$-restriction estimate and that the pair $(S, Sigma)$ meets a sharp smallness condition, it yields explicit annihilation constants $A_{ ext{ann}}(S, Sigma)$, strengthening prior UP results. The work specializes to finite abelian groups, delivering probabilistic constructions with random spectra and deterministic Λ_q-based criteria, plus deterministic energy-based bounds. In the Euclidean setting $G= R^d$, restriction theory for spheres and their neighborhoods yields concrete, near-optimal annihilation constants for time-frequency concentration problems, including constructions where large spatial and spectral supports still admit a fixed annihilation constant. Collectively, the results provide a versatile, quantitative toolkit for time–frequency localization in discrete and continuous groups, with implications for signal processing and control theory.
Abstract
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 $Σ\subset \widehat{G}$, there exists a constant $C(S, Σ)$ such that for any $f \in L^2(G)$, the following inequality holds: \[\|f\|_{L^2(G)} \leq C(S, Σ) \bigl( \|f\|_{L^2(G \setminus S)} + \|\widehat{f}\|_{L^2(\widehat{G} \setminus Σ)} \bigr),\] 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 $Σ$ satisfies a restriction theorem and to provide an estimate for the constant $C(S, Σ)$. 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.