  We prove that the Fourier transform of self-similar measures on the complex
plane has fast decay outside of a very sparse set of frequencies, with
quantitative estimates, extending the results obtained in the real line, first
by R. Kaufman, and later, with quantitative bounds, by the first author and P.
Shmerkin. Also we derive several applications concerning correlation dimension
and Frostman exponent of these measures. Furthermore, we present a
generalization for a particular case on $\mathbb{R}^n,$ with $n\ge3.$
