# 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 $R$ associated to contractions $f_{j}(x)=r_{j}x+b_{j}$, $j∈A$, for some finite $A$, such that $F$ is not a singleton. We prove that if $gr_{i}/gr_{j}$ is irrational for some $i=j$, 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 $μ$ on $F$ is a Rajchman measure: the Fourier transform $μ (ξ)→0$ as $∣ξ∣→∞$. The rate of $μ (ξ)→0$ is also shown to be logarithmic if $gr_{i}/gr_{j}$ is diophantine for some $i=j$. The proof is based on quantitative renewal theorems for stopping times of random walks on $R$.

## 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