On fine fractal properties of generalized infinite Bernoulli convolutions

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions mu(xi) of the following random variables: xi = Sigma(infinity)(k=1) xi(k)a(k). where a(k) are terms of a given positive convergent series; xi(k) are independent random variables taking values 0 and 1 with probabilities p(0k) and p(1k) correspondingly. We give (without any restriction on {a(n})) necessary and sufficient conditions for the topological support of xi to be a nowhere dense set. Fractal properties of the topological support of xi and fine fractal properties of the corresponding probability measure mu(xi) itself are studied in details for the case where a(k) >= r(k) : = a(k+1) + a(k+2) + ... (i.e., r(k-1) >= 2r(k)) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff-Besicovitch dimension) supports of mu(xi) for the above mentioned case is also studied in details. We describe a series of sets (with additional structural properties) which play the role of minimal dimensional supports of generalized Bernoulli convolutions. We also show how a generalization of M. Cooper's dimensional results on symmetric Bernoulli convolutions can easily be derived from our results. (c) 2008 Elsevier Masson SAS. All rights reserved.