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We study properties of Bernoulli convolutions generated by the secondOstrogradsky 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 $q_{k+1}\geqq_k(q_k+1)$, and $\{\xi_k\}$ are independent random variables taking the values$0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We provethat $\xi$ has an anomalously fractal Cantor type singular distribution($\dim_H (S_{\xi})=0$) whose Fourier-Stieltjes transform does not tend to zeroat infinity. We also develop different approaches how to estimate a level of"irregularity" of probability distributions whose spectra are of zero Hausdorffdimension. Using generalizations of the Hausdorff measures and dimensions, finefractal properties of the probability measure $\mu_\xi$ are studied in details.Conditions for the Hausdorff--Billingsley dimension preservation on thespectrum by its probability distribution function are also obtained.

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