Fourier decay of self-similar measures on the complex plane

TL;DR

This paper demonstrates that the Fourier transform of self-similar measures on the complex plane decays rapidly outside sparse frequency sets, extending real-line results and exploring implications for measure dimensions.

Contribution

It extends Fourier decay results of self-similar measures from the real line to the complex plane and generalizes some findings to higher dimensions.

Findings

Fourier transform exhibits rapid decay outside sparse frequencies.

Applications to correlation dimension and Frostman exponent.

Generalization to certain cases in higher dimensions (ℝⁿ, n≥3).

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. 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 $n\ge3.$