On the dimension of Bernoulli convolutions.

The Bernoulli convolution with parameter ..(0,1) is the probability measure .. that is the law of the random variable .n.0±.n , where the signs are independent unbiased coin tosses. We prove that each parameter ..(1/2,1) with dim..<1 can be approximated by algebraic parameters ..(1/2,1) within an error of order exp(.deg(.)A) such that dim..<1, for any number A. As a corollary, we conclude that dim..=1 for each of .=ln2,e.1/2,./4. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer.s conjecture implies the existence of a constant a<1 such that dim..=1 for all ..(a,1).