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 $\delta$-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.

44 pages, arguments simplified and 2 figures added

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Cited by in corpus (10)

• An introduction to fractal uncertainty principle
• Semiclassical measures for higher dimensional quantum cat maps
• Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle
• Fractal Uncertainty Principle with Explicit Exponent
• A generalization of the Beurling--Malliavin Majorant theorem
• Inverse theorems for discretized sums and $L^q$ norms of convolutions in ${\mathbb{R}}^d$
• Analyticity and observability for fractional heat equation on ${\mathbb{R}}^n$
• The fractal uncertainty principle via Dolgopyat's method in higher dimensions
• The Beurling and Malliavin Theorem in Several Dimensions
• Fractal uncertainty principle for discrete Cantor sets with random alphabets