Multifractal structure of Bernoulli convolutions.

Let νpλ be the distribution of the random series $\sum_{n=1}^\infty i_n \lam^n$, where in 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 νpλ for typical λ. Namely, we investigate the size of the sets \begin{linenomath} \[ \Delta_{\lam, p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log \nula^p(B(x, r))}{\log r} =\alpha\right\}\!. \]\end{linenomath} Our main results highlight the fact that for almost all, and in some cases all, λ in an appropriate range, Δλ, p(α) is nonempty and, moreover, has positive Hausdorff dimension, for many values of α. This happens even in parameter regions for which νpλ is typically absolutely continuous. [ABSTRACT FROM PUBLISHER]