Fourier decay bound and differential images of self-similar measures

TL;DR

This paper proves that certain differential images of self-similar measures exhibit power Fourier decay, extending known results and applying combinatorial methods to fractal analysis and normal number existence.

Contribution

It establishes Fourier decay bounds for $C^2$ differential images of self-similar measures under broad conditions, extending Kaufman's results on Bernoulli convolutions.

Findings

Fourier transforms of differential images decay polynomially.

The decay holds for all contraction ratios in (0, 1/m).

Application to normal numbers in fractals.

Abstract

In this note, we investigate $C^2$ differential images of the homogeneous self-similar measure associated with an IFS $\mathcal{I}=\{\rho x+a_j\}_{j=1}^m$ satisfying the strong separation condition and a positive probability vector $\vec{p}$. It is shown that the Fourier transforms of such image measures have power decay for any contractive ratio $\rho\in (0, 1/m)$, any translation vector $\vec{a}=(a_1, \ldots, a_m)$ and probability vector $\vec{p}$, which extends a result of Kaufman on Bernoulli convolutions. Our proof relies on a key combinatorial lemma originated from Erd\H{o}s, which is important in estimating the oscillatory integrals. An application to the existence of normal numbers in fractals is also given.