Abstract
Let $____nu_____lambda^p$ be the distribution of the random series $____sum_{n=1}^____infty i_n ____lambda^n$, 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 $____nu_____lambda^p$ for typical $____lambda$. Namely, we investigate the size of the sets ____[ ____Delta_{____lambda,p}(____alpha) = ____left____{x____in____R: ____lim_{r____searrow 0} ____frac{____log ____nu_____lambda^p(B(x,r))}{____log r} =____alpha____right____}. ____] 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 $____nu_____lambda^p$ is typically absolutely continuous.