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

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.