Multifractal structure of Bernoulli convolutions

Let. p. be the distribution of the random series Sigma(infinity)(n=1) in lambda(n), where i(n) is a sequence of i.i.d. random variables taking the values 0,1 with probabilities p, 1 - p. These measures are the well-known (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of nu(p)(lambda) for typical lambda. Namely, we investigate the size of the sets Delta(lambda, p) (alpha) = {x is an element of R : lim(r SE arrow 0) log nu(p)(lambda) (B(x,r))/log r = alpha}. Our main results highlight the fact that for almost all, and in some cases all, lambda in an appropriate range, Delta(lambda, p) (alpha) is nonempty and, moreover, has positive Hausdorff dimension, for many values of alpha. This happens even in parameter regions for which nu(p)(lambda) is typically absolutely continuous.