  We establish a version of the fractal uncertainty principle, obtained by
Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is
limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many
products of $\delta$-regular sets in one dimension, but relative to arbitrary
axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our
method combines the original approach by Bourgain and Dyatlov, in the more
quantitative 2017 rendition by Jin and Zhang, with Cartan set techniques.
