Mathematical Research Letters 3, 231–239 (1996) ABSOLUTE CONTINUITY OF BERNOULLI CONVOLUTIONS, A SIMPLE PROOF Yuval Peres and Boris Solomyak Abstract. The distribution ν λ of the random series ∑ ±λ n has been studied by many authors since the two seminal papers by Erd˝os in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urba´ nski, 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 ν λ is absolutely continuous for a.e. λ ∈ (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. 1. Introduction Consider the random series Y λ = ∞  n=0 ±λ n , for 0 <λ< 1, where the “+” and “-” signs are chosen independently with probability 1/2. Let ν λ be the distribution of Y λ : ν λ (E) = Prob{Y λ ∈ E}. The measure ν λ is the infinite convolution product of 1 2 (δ -λ n + δ λ n ) for n ≥ 0 and is sometimes called an “infinite Bernoulli convolution”. The Fourier transform of ν λ can be represented as a convergent infinite product: ˆ ν λ (u)=  ∞ n=0 cos(λ n u). Infinite Bernoulli convolutions have been studied since the 1930’s; we sketch some of the relevant background below. The following theorem was recently proved by the second author, who verified a conjecture of Garsia (1962). Our goal here is to present a self-contained simple proof of this result. Received December 10, 1995. 1991 Mathematics Subject Classification. Primary 28A12, Secondary 42A61, 60G50. The authors were supported in part by NSF grants DMS–9404391 and DMS–9500744. 231