Multifractal analysis of Bernoulli convolutions associated with Salem numbers

We consider the multifractal structure of the Bernoulli convolution nu(lambda), where lambda(-1) is a Salem number in (1, 2). Let tau(q) denote the L-q-spectrum of nu(lambda). We show that if alpha is an element of [tau'(+infinity), tau'/(0+)], then the level set E(alpha) := {x is an element of R: lim(r -> 0) log nu(lambda)([x - r, x + r])/log r = alpha} is non-empty and dim(H) E(alpha) = tau*(alpha), where tau* denotes the Legendre transform of tau. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval [tau'(+infinity), tau'(0+)] is not a singleton when lambda(-1) is the largest real root of the polynomial x(n) - x(n-1) - ... - x + 1, n >= 4. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures. (C) 2011 Elsevier Inc. All rights reserved.