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.