Absolute Continuity of Bernoulli Convolutions, A Simple Proof

The distribution $\nu_\lambda$ of the random series $\sum \pm \lambda^n$ has been studied by many authors since the two seminal papers by Erd\H{o}s in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urba\'{n}ski, and Ledrappier showed the importance of these distributions in several problems in dynamical systems and Hausdorff dimension estimation. Recently the second author proved a conjecture made by Garsia in 1962, that $\nu_\lambda$ is absolutely continuous for a.e.\ $\lambda \in (1/2,1)$. Here we give a considerably simplified proof of this theorem, using differentiation of measures instead of Fourier transform methods. This technique is better suited to analyze more general random power series.

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Published 1 January 1996