A higher dimensional Bourgain-Dyatlov fractal uncertainty principle

TL;DR

This paper extends the fractal uncertainty principle to higher dimensions, allowing Fourier support on complex sets covered by products of regular sets, and remains valid under distortions, combining previous methods with Cartan set techniques.

Contribution

It generalizes the Bourgain-Dyatlov fractal uncertainty principle to higher dimensions and more flexible sets, using a novel combination of existing approaches and Cartan set techniques.

Findings

Established higher-dimensional fractal uncertainty principle.

Applicable to sets covered by products of regular sets in arbitrary axes.

Results hold under diffeomorphic distortions.

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 $\delta$-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.