  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 $\mathbb{R}$.
