Fractal properties of some Bernoulli convolutions

We study the structure and topological, metric, and fractal properties of the distribution of the random variable \[ \xi =\sum _{k=1}^\infty \xi _k a_k, \] where $\sum _{k=1}^\infty a_k$ is a convergent series of positive terms $a_k$ such that \[ a_{3k-2}= a_{3k-1}+a_{3k}, \] $a_{i} \ge a_{i+1}+a_{i+2}+\cdots$, $i \ne 3k-2$, $k \in \mathbf N$, and where $\xi _k$ are independent random variables assuming two values $0$ and $1$ with the probabilities $p_{0k}$ and $p_{1k}$, respectively. We prove that the distribution of $\xi$ is either purely discrete or purely singularly continuous. We obtain the criteria for a distribution to belong to each of these types. The topological-metric structure of the distribution is studied in the continuous case. The main result of the paper describes the fractal properties of the distribution of the random variable $\xi$. The relations are obtained for the Hausdorff–Besicovitch spectrum (the minimal closed support) of the random variable $\xi$ and for the Hausdorff dimension of the corresponding probability measure $\mu _{\xi }$.