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

Abstract

Let $\hat{I}»\in (1,\sqrt{2}]$ be an algebraic integer with Mahler measure $2.$ A classical result of Garsia shows that the Bernoulli convolution $\hat{I}{¼}_{\hat{I}}»$ 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.

Version accepted in Proc. Am. Math. Soc