Recent progress on Bernoulli convolutions

TL;DR

This paper surveys recent advances in understanding the regularity properties of Bernoulli convolutions, which are probability measures generated by random power series with parameters in (0,1).

Contribution

It provides a comprehensive overview of recent progress in analyzing the regularity of Bernoulli convolutions, highlighting new theoretical insights.

Findings

Improved understanding of measure regularity for certain parameters

Identification of conditions for absolute continuity

Connections between algebraic properties and measure smoothness

Abstract

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the measure on $\bf R$ that is the distribution of the random power series $\sum\pm\lambda^n$, where $\pm$ are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.