Fractal uncertainty principle with explicit exponent

Abstract

We prove an explicit formula for the dependence of the exponent $$\beta $$ in the fractal uncertainty principle of Bourgain–Dyatlov (Ann Math 187:1–43, 2018) on the dimension $$\delta $$ and on the regularity constant $$C_R$$ for the regular set. In particular, this implies an explicit essential spectral gap for convex co-compact hyperbolic surfaces when the Hausdorff dimension of the limit set is close to 1.