The Fourier Ratio: Uncertainty, Restriction, and Approximation for Compactly Supported Measures

Abstract: We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure (μ) on (\mathbb{R}^d) and (f\in L^2(μ)), the Fourier ratio compares (L^1) and (L^2) norms of a regularized Fourier transform at scale (R). We develop a fractal uncertainty principle giving sharp two-sided bounds in terms of covering numbers of spatial and frequency supports, with applications to exact signal recovery. We show that small Fourier ratio implies efficient approximation by low-degree trigonometric polynomials in (L^1), (L^2), and (L^\infty). In contrast, restriction estimates reveal a sharp gap between curved measures and random fractal measures, yielding strong lower bounds on approximation degree. Applications to convex surface measures are also obtained.