  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.
