A higher dimensional Bourgain-Dyatlov fractal uncertainty principle

Abstract

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 $\hat{I}´$-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.

44 pages, arguments simplified and 2 figures added