Quantitative FUP and spectral gap for quasi-Fuchsian group

We derive an explicit formula for the exponent $β$ in the higher-dimensional fractal uncertainty principle (FUP) established by Cohen 2023, quantifying its dependence on the porosity parameter $ν$ 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.