On Bernoulli convolutions generated by second Ostrogradsky series and their fine fractal properties

Abstract: We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} \xi = \sum_{k=1}^\infty \frac{(-1)^{k+1}\xi_k}{q_k}, \end{equation} where $q_k$ is a sequence of positive integers with $q_{k+1}\geq q_k(q_k+1)$, and ${\xi_k}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $\xi$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_{\xi})=0$) whose Fourier-Stieltjes transform does not tend to zero at infinity. We also develop different approaches how to estimate a level of "irregularity" of probability distributions whose spectra are of zero Hausdorff dimension. Using generalizations of the Hausdorff measures and dimensions, fine fractal properties of the probability measure $\mu_\xi$ are studied in details. Conditions for the Hausdorff--Billingsley dimension preservation on the spectrum by its probability distribution function are also obtained.