Quantitative FUP and spectral gap for quasi-Fuchsian group

TL;DR

This paper provides an explicit formula for the spectral gap in higher-dimensional fractal uncertainty principles, linking it to the porosity of Fourier support, and applies it to convex co-compact hyperbolic 3-manifolds.

Contribution

It introduces a quantitative version of the fractal uncertainty principle with an explicit spectral gap formula, extending previous results to higher dimensions.

Findings

Explicit formula for exponent β in FUP based on porosity ν

Refined spectral gap for convex co-compact hyperbolic 3-manifolds

Extension of Jin-Zhang 2020 to higher dimensions

Abstract

We derive an explicit formula for the exponent $\beta$ in the higher-dimensional fractal uncertainty principle (FUP) established by Cohen 2023, quantifying its dependence on the porosity parameter $\nu$ of the Fourier support. This quantitative version of FUP yields an explicit essential spectral gap for convex co-compact hyperbolic 3-manifolds arising from quasi-Fuchsian groups, thereby refining the result of Tao 2025. Our result extends the earlier work of Jin-Zhang 2020 to higher dimensions.