The equivalence of some Bernoulli convolutions to Lebesgue measure

Since the 1930's many authors have studied the distribution $\nu _{\lambda}$ of the random series $Y_{\lambda}=\sum \pm {\lambda}^n$ where the signs are chosen independently with probability $(1/2,1/2)$ and $0<\lambda<1$. Solomyak recently proved that for almost every $\lambda\in [\frac{1}{2},1],$ the distribution $\nu _{\lambda}$ is absolutely continuous with respect to Lebesgue measure. In this paper we prove that $\nu _{\lambda}$ is even equivalent to Lebesgue measure for almost all $\lambda\in [\frac{1}{2},1]$.