# Trigonometric series and self-similar sets

### Jialun Li

Universität Zürich, Switzerland### Tuomas Sahlsten

University of Manchester, UK

## Abstract

Let ${F}$ be a self-similar set on $\mathbb{R}$ associated to contractions ${f_j(x) = r_j x + b_j}$, ${j \in \mathcal{A}}$, for some finite $\mathcal{A}$, such that ${F}$ is not a singleton. We prove that if ${\log r_i / {\log r_j}}$ is irrational for some ${i \neq j}$, then ${F}$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of ${F}$. No separation conditions are assumed on ${F}$. We establish our result by showing that every self-similar measure ${\mu}$ on ${F}$ is a Rajchman measure: the Fourier transform ${\widehat{\mu}(\xi) \to 0}$ as ${|\xi| \to \infty}$. The rate of ${\widehat{\mu}(\xi) \to 0}$ is also shown to be logarithmic if ${\log r_i / {\log r_j}}$ is diophantine for some ${i \neq j}$. The proof is based on quantitative renewal theorems for stopping times of random walks on $\mathbb{R}$.

## Cite this article

Jialun Li, Tuomas Sahlsten, Trigonometric series and self-similar sets. J. Eur. Math. Soc. 24 (2022), no. 1, pp. 341–368

DOI 10.4171/JEMS/1102