The fractal uncertainty principle via Dolgopyat's method in higher dimensions

We prove a fractal uncertainty principle with exponent $\frac{d}{2}-\delta +\epsilon$, $\epsilon >0$, for Ahlfors–David regular subsets of ${\mathbb{R}}^d$ with dimension $\delta$ which satisfy a suitable “nonorthogonality condition”. This generalizes the application of Dolgopyat’s method by Dyatlov–Jin (arXiv:1702.03619) to prove the same result in the special case $d=1$. As a corollary, we get a quantitative spectral gap for the Laplacian on convex cocompact hyperbolic manifolds of arbitrary dimension with Zariski dense fundamental groups.