# Torbin H. M.

### Singularity and fine fractal properties of one class of generalized infinite Bernoulli convolutions with essential overlaps. II

Ukr. Mat. Zh. - 2015. - 67, № 12. - pp. 1667-1678

We discuss the Lebesgue structure and fine fractal properties of infinite Bernoulli convolutions, i.e., the distributions of random variables $\xi=\sum_{k=1}^{\infty}\xi_ka_k$, where $\sum_{k=1}^{\infty}a_k$ is a convergent positive series and $\xi_k$ are independent (generally
speaking, nonidentically distributed) Bernoulli random variables. Our main aim is to investigate the class of Bernoulli convolutions with essential overlaps generated by a series $\sum_{k=1}^{\infty}a_k$, such that, for any $k\in \mathbb{N}$, there exists $s_k\in \mathbb{N}\cup\{0\}$ for which $a_k = a_{k+1} = . . . = a_{k+s_k} ≥ r_{k+s_k}$ and, in addition, $s_k > 0$ for infinitely many indices $k$. In this case, almost
all (both in a sense of Lebesgue measure and in a sense of fractal dimension) points from the spectrum have continuum many representations of the form $\xi=\sum_{k=1}^{\infty}\varepsilon_ka_k$, with $\varepsilon_k\in\{0, 1\}$. It is proved that $\mu_\xi$ has either a pure discrete distribution
or a pure singulary continuous distribution.

We also establish sufficient conditions for the faithfulness of the family of cylindrical intervals on the spectrum $\mu_\xi$
generated by the distributions of the random variables $\xi$. In the case of singularity, we also deduce the explicit formula
for the Hausdorff dimension of the corresponding probability measure [i.e., the Hausdorff–Besicovitch dimension of the
minimal supports of the measure $\mu_\xi$ (in a sense of dimension)].

### Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series

Baranovskyi O. M., Pratsiovytyi M. V., Torbin H. M.

Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1155–1168

We study topological and metric properties of the set $$C\left[\overline{O}^1, \{V_n\}\right] = \left\{x:\; x= ∑_n \frac{(−1)^{n−1}}{g_1(g_1 + g_2)…(g_1 + g_2 + … + g_n)},\quad g_k ∈ V_k ⊂ \mathbb{N}\right\}$$ with certain conditions on the sequence of sets $\{V_n\}$. In particular, we establish conditions under which the Lebesgue measure of this set is (a) zero and (b) positive. We compare the results obtained with the corresponding results for continued fractions and discuss their possible applications to probability theory.

### Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits

Pratsiovytyi M. V., Torbin H. M.

Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1163–1170

Dedicated to V. S. Korolyuk on occasion of his 80-th birthday

Properties of the set $T_s$ of "particularly nonnormal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose $s$-adic digits have the asymptotic frequencies in the nonterminating $s$-adic expansion of $x$, and some do not).
It is proven that the set $T_s$ is residual in the topological sense (i.e., it is of the first Baire category)
and it is generic in the sense of fractal geometry ( $T_s$ is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1).
A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their $s$-adic expansions is presented.

### Multifractal Analysis of Singularly Continuous Probability Measures

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 706–720

We analyze correlations between different approaches to the definition of the Hausdorff dimension of singular probability measures on the basis of fractal analysis of essential supports of these measures. We introduce characteristic multifractal measures of the first and higher orders. Using these measures, we carry out the multifractal analysis of singular probability measures and prove theorems on the structural representation of these measures.

### Superfractality of the set of numbers having no frequency of*n*-adic digits, and fractal probability distributions

Pratsiovytyi M. V., Torbin H. M.

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 971–975

We study the fractal properties (we find the Hausdorff-Bezikovich dimension and Hausdorff measure) of the spectrum of a random variable with independent*n*-adic (*n*≥2,*n* ∃*N* digits, the infinite set of which is fixed. We prove that the set of numbers of the segment [0, 1] that have no frequency of at least one*n*-adic digit is superfractal.