Fourier decay of self-similar measures and self-similar sets of uniqueness

Varjú, Péter P.; Yu, Han

doi:10.2140/apde.2022.15.843

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Fourier decay of self-similar measures and self-similar sets of uniqueness

Péter P. Varjú and Han Yu

Vol. 15 (2022), No. 3, 843–858

DOI: 10.2140/apde.2022.15.843

Abstract

We investigate the Fourier transform of self-similar measures on $ℝ$ . We provide quantitative decay rates of Fourier transform of some self-similar measures. Our method is based on random walks on lattices and Diophantine approximation in number fields. We also completely identify all self-similar sets which are sets of uniqueness. This generalizes a classical result of Salem and Zygmund.

Keywords

Rajchman measure, Fourier decay of self-similar measures, sets of uniqueness, digit changes

Mathematical Subject Classification

Primary: 11A63, 28A80, 42A16

Milestones

Received: 4 May 2020

Revised: 2 September 2020

Accepted: 27 October 2020

Published: 10 June 2022

Authors

Péter P. Varjú
Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

Han Yu
Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

Vol. 15, No. 3, 2022