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

We prove a fractal uncertainty principle with exponent \(\frac{d}{2} - \delta + \varepsilon\), \(\varepsilon > 0\), 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.