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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 $|\\xi| \\to \\infty$. 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