A higher-dimensional Bourgain–Dyatlov fractal uncertainty principle

Han, Rui; Schlag, Wilhelm

doi:10.2140/apde.2020.13.813

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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 $δ$ -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.

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

Vol. 13, No. 3, 2020