ON THE DIMENSION OF BERNOULLI CONVOLUTIONS

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