A higher dimensional Bourgain-Dyatlov fractal uncertainty principle

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.