Absolute Continuity of Self Similar Measures

TL;DR

This paper establishes conditions under which self-similar measures, including Bernoulli convolutions, are absolutely continuous, linking entropy, contraction ratios, and Mahler measure to measure regularity.

Contribution

It provides a new criterion involving Garsia entropy and Mahler measure for the absolute continuity of self-similar measures, including Bernoulli convolutions.

Findings

Self-similar measures are absolutely continuous under specific entropy and separation conditions.

A sufficient condition for Bernoulli convolutions to be absolutely continuous is derived.

The results connect measure regularity with algebraic properties of parameters.

Abstract

We prove that a self similar measure is absolutely continuous providing that it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations of the self similar measure. In the special case of Bernoulli convolutions this paper gives a sufficient condition for the Bernoulli convolution with parameter lambda to be absolutely continuous in terms of lambda and its Mahler measure.