We establish a complete algebraic characterization of self-similar iterated function systems $\Phi$ on $\mathbb{R}^{d}$, for which there exists a positive probability vector $p$ so that the Fourier transform of the self-similar measure corresponding to $\Phi$ and $p$ does not tend to $0$ at infinity.
On the Fourier transforms of self-similar measures
Polynomial approximation of self-similar measures and the spectrum of the transfer operator
On multifractal formalism for self-similar measures with overlaps
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