Trigonometric series and self-similar sets

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 $i\ne j$, 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 $\hat{\mu }(\xi )\to 0$ as $|\xi |\to \infty$. The rate of $\hat{\mu }(\xi )\to 0$ is also shown to be logarithmic if $\log r_i/\log r_j$ is diophantine for some $i\ne j$. The proof is based on quantitative renewal theorems for stopping times of random walks on $\mathbb{R}$.