The equivalence of some Bernoulli convolutions to Lebesgue measure

Abstract:

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 $\lambda \in [\frac {1}{2},1]$.