Towards Absolutely Continuous Bernoulli Convolutions

TL;DR

This paper links the absolute continuity of Bernoulli convolutions to counting overlaps and uses ergodic theory for hyperbolic algebraic integers to analyze this property.

Contribution

It introduces a novel approach connecting Bernoulli convolutions' absolute continuity to overlap counting and ergodic theory for algebraic integers.

Findings

Established a method to analyze absolute continuity via overlap growth

Connected Bernoulli convolutions to ergodic theory of cocycles

Provided new insights for hyperbolic algebraic integers

Abstract

We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.