  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.
