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
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
Milestones
Received: 30 August 2018
Revised: 8 February 2019
Accepted: 20 March 2019
Published: 15 April 2020
Authors
 Rui Han School of Mathematics Georgia Institute of Technology Atlanta, GA United States Wilhelm Schlag Department of Mathematics Yale University New Haven, CT United States