Self-similar measures: asymptotic bounds for the dimension and Fourier decay of smooth images

TL;DR

This paper extends results on Fourier decay of self-similar measures to homogeneous cases, providing quantitative bounds and applications to dimension growth, Fourier decay of smooth images, and Bernoulli convolutions.

Contribution

It offers a quantitative version of Fourier decay for homogeneous self-similar measures and explores their implications for dimensions and Fourier properties.

Findings

Fourier transform of homogeneous self-similar measures exhibits power decay with explicit bounds.

Convolving with such measures increases correlation dimension quantitatively.

Dimension and Frostman exponent of Bernoulli convolutions approach 1 as contraction ratio approaches 1.

Abstract

R. Kaufman and M. Tsujii proved that the Fourier transform of self-similar measures has a power decay outside of a sparse set of frequencies. We present a version of this result for homogeneous self-similar measures, with quantitative estimates, and derive several applications: (1) non-linear smooth images of homogeneous self-similar measures have a power Fourier decay, (2) convolving with a homogeneous self-similar measure increases correlation dimension by a quantitative amount, (3) the dimension and Frostman exponent of (biased) Bernoulli convolutions tend to $1$ as the contraction ratio tends to $1$, at an explicit quantitative rate.