Vol. 15, No. 3, 2022

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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
Abstract

We investigate the Fourier transform of self-similar measures on . 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