Mathematical Research Letters

Volume 3 (1996)

Number 2

Absolute Continuity of Bernoulli Convolutions, A Simple Proof

Pages: 231 – 239

DOI: http://dx.doi.org/10.4310/MRL.1996.v3.n2.a8

Authors

Yuval Peres (University of California at Berkeley)

Boris Solomyak (University of Washington, Seattle)

Abstract

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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