#### Vol. 13, No. 3, 2020

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A higher-dimensional Bourgain–Dyatlov fractal uncertainty principle

### Rui Han and Wilhelm Schlag

Vol. 13 (2020), No. 3, 813–863
DOI: 10.2140/apde.2020.13.813
##### 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 $Y\subset {ℝ}^{d}$ 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.

##### Keywords
uncertainty principle, fractal sets, subharmonic functions, Cartan estimates, Beurling–Malliavin theorem
##### Mathematical Subject Classification 2010
Primary: 32U15, 42B30