Absolute continuity of Bernoulli convolutions, a simple proof

The distribution νλ">νλνλ of the random series ∑±λn">∑±λn∑±λn 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 νλ">νλνλ is absolutely continuous for a.e.\ λ∈(1/2,1)">λ∈(1/2,1)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.