Bernoulli convolutions for algebraic parameters

The Bernoulli convolution with parameter lambda is the law of the random variable: Sum X_i lambda^i, where X_i are independent unbiased +1/-1 valued random variables. If lambda <1/2, then the Bernoulli convolution is singular and is supported on a Cantor set. If 1> lambda >1/2, the question whether the Bernoulli convolution is singular or a.c. is a very interesting open problem. Recent papers of Hochman and Shmerkin prove that the set of lambda's such that the measure is singular is of Hausdorff dimension 0. I will discuss the problem for parameters lambda that are algebraic. Work in progress, joint with Emmanuel Breuillard.

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