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 d which can be covered by finitely many products of δ-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.

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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