Absolute continuity of Bernoulli convolutions for algebraic parameters

TL;DR

This paper proves that Bernoulli convolutions are absolutely continuous when the parameter is an algebraic number near 1, with the closeness depending on its Mahler measure, advancing understanding of their measure-theoretic properties.

Contribution

It establishes absolute continuity of Bernoulli convolutions for algebraic parameters close to 1, linking the property to the Mahler measure of the parameter.

Findings

Bernoulli convolutions are absolutely continuous for certain algebraic parameters near 1.

The closeness to 1 depends on the Mahler measure of the algebraic parameter.

Provides conditions under which Bernoulli convolutions have a smooth density.

Abstract

We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.