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

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure $\mu$ on $\mathbb{R}^d$ and $f\in L^2(\mu)$, 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.