Absolute continuity of bernoulli convolutions, a simple proof

The distribution νλ of the random series Σ ±λⁿ has been studied by many authors since the two seminal papers by Erdo″s in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urbański, 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 νλ 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.