On fine fractal properties of generalized infinite Bernoulli convolutions

Abstract

The paper is devoted to the investigation of generalized infinite Bernoulli convolutions, i.e., the distributions μ ξ of the following random variables: ξ = ∑ k = 1 ∞ ξ k a k , where a k are terms of a given positive convergent series; ξ k are independent random variables taking values 0 and 1 with probabilities p 0 k and p 1 k correspondingly. We give (without any restriction on { a n } ) necessary and sufficient conditions for the topological support of ξ to be a nowhere dense set. Fractal properties of the topological support of ξ and fine fractal properties of the corresponding probability measure μ ξ itself are studied in details for the case where a k ⩾ r k : = a k + 1 + a k + 2 + ⋯ (i.e., r k − 1 ⩾ 2 r k ) for all sufficiently large k. The family of minimal dimensional (in the sense of the Hausdorff–Besicovitch dimension) supports of μ ξ 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.