Fourier decay behavior of homogeneous self-similar measures on the complex plane

Abstract

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. We also derive several applications concerning correlation dimension and Frostman exponent of complex Bernoulli convolutions. Furthermore, we present a generalization for a particular case on ${\mathbb{R}}^d,$ with $d\ge 3$.