Bernoulli convolutions with Garsia parameters in $(1,\sqrt{2}]$ have continuous density functions

TL;DR

This paper proves that for certain algebraic integers in the interval (1,√2], the density functions of their associated Bernoulli convolutions are not only bounded but also continuous, extending classical results.

Contribution

It establishes the continuity of the density functions for Bernoulli convolutions with Garsia parameters in (1,√2], building on Garsia's classical absolute continuity result.

Findings

Density functions are continuous for the specified parameters.

Extends Garsia's classical result from boundedness to continuity.

Focuses on algebraic integers with Mahler measure 2 in (1,√2].

Abstract

Let $\lambda\in (1,\sqrt{2}]$ be an algebraic integer with Mahler measure $2.$ A classical result of Garsia shows that the Bernoulli convolution $\mu_\lambda$ is absolutely continuous with respect to the Lebesgue measure with a density function in $L^\infty$. In this paper, we show that the density function is continuous.