The fractal uncertainty principle via Dolgopyat’smethod in higher dimensions

We prove a fractal uncertainty principle with exponent 1 2 d + , > 0, for Ahlfors-David regular subsets of d with dimension which satisfy a suitable "nonorthogonality condition".This generalizes the application of Dolgopyat's method by Dyatlov and Jin (2018) to higher dimensions.As a corollary, we get a quantitative essential spectral gap for the Laplacian on convex cocompact hyperbolic manifolds of arbitrary dimension with Zariski-dense fundamental groups.