On the Rajchman property for self-similar measures on $\mathbb{R}^{d}$

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.