Absolute continuity of non-homogeneous self-similar measures

TL;DR

This paper proves that non-homogeneous self-similar measures on the real line are absolutely continuous for most parameters, advancing understanding beyond homogeneous cases and confirming a longstanding conjecture.

Contribution

It introduces new methods to establish absolute continuity in the non-homogeneous setting, surpassing classical transversality techniques.

Findings

Almost all parameters in the super-critical region yield absolutely continuous measures.

New results on the dimension and Fourier decay of certain random self-similar measures.

First significant improvement over classical methods for non-homogeneous self-similar measures.

Abstract

We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S-M. Ngai and Y. Wang. While recently there has been much progress in understanding absolute continuity for homogeneous self-similar measures, this is the first improvement over the classical transversality method in the general (non-homogeneous) case. In the course of the proof, we establish new results on the dimension and Fourier decay of a class of random self-similar measures.