# Pratsiovytyi M. V.

### Superfractality of the set of incomplete sums of one positive series

Markitan V. P., Pratsiovytyi M. V., Savchenko I. O.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 10. - pp. 1403-1416

We consider a family of convergent positive normed series with real terms defined by the conditions $$\sum ^{\infty}_{n=1} d_n = \underbrace{c_1 + ...+c_1}_{a_1} + \underbrace{c_2 + ...+c_2}_{a_2} + ... + \underbrace{c_n + ...+c_n}_{a_n} + \widetilde{ r_n} = 1,$$ where $(a_n)$ is a nondecreasing sequence of real numbers. The structural properties of these series are investigated. For a partial case, namely, $(a_n) = 2^{n - 1}, c_n = (n + 1)\widetilde {r_n}, n \in N$, we study the geometry of the series (i.e., the properties of cylindrical sets, metric relations generated by them, and topological and metric properties of the set of all incomplete sums of the series). For the infinite Bernoulli convolution determined we describe its Lebesgue structure (discrete, absolutely continuous, and singular components) and spectral properties, as well as the behavior of the absolute value of the characteristic function at infinity. We also study the finite autoconvolutions of distributions of this kind.

### Frequency of a Digit in the Representation of a Number and the Asymptotic Mean Value of the Digits

Klymchuk S. O., Makarchuk O. P., Pratsiovytyi M. V.

Ukr. Mat. Zh. - 2014. - 66, № 3. - pp. 302–310

We study the relationship between the frequency of a ternary digit in a number and the asymptotic mean value of the digits. The conditions for the existence of the asymptotic mean of digits in a ternary number are established. We indicate an infinite everywhere dense set of numbers without frequency of digits but with the asymptotic mean of the digits.

### Distribution of Random Variable Represented by a Binary Fraction with Three Identically Distributed Redundant Digits

Makarchuk O. P., Pratsiovytyi M. V.

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 79–88

We present the complete solution of the problem of pure Lebesgue type of the distribution of random variable *χ* represented by a binary fraction with three identically distributed redundant digits.

### Self-Affine Singular and Nowhere Monotone Functions Related to the *Q*-Representation of Real Numbers

Kalashnikov A. V., Pratsiovytyi M. V.

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 405-417

We study functional, differential, integral, self-affine, and fractal properties of continuous functions belonging to a finite-parameter family of functions with a continuum set of "peculiarities". Almost all functions of this family are singular (their derivative is equal to zero almost everywhere in the sense of Lebesgue) or nowhere monotone, in particular, nondifferentiable. We consider different approaches to the definition of these functions (using a system of functional equations, projectors of symbols of different representations, distribution of random variables, etc.).

### Mykola Ivanovych Shkil' (on his 80th birthday)

Korolyuk V. S., Lukovsky I. O., Perestyuk N. A., Pratsiovytyi M. V., Samoilenko A. M., Yakovets V. P.

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1720-1722

### $A_2$-continued fraction representation of real numbers and its geometry

Dmytrenko S. O., Kyurchev D. V., Pratsiovytyi M. V.

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 452-463

We study the geometry of representations of numbers by continued fractions whose elements belong to the set $A_2 = {α_1, α_2}$ ($A_2$-continued fraction representation). It is shown that, for $α_1 α_2 ≤ 1/2$, every point of a certain segment admits an $A_2$-continued fraction representation. Moreover, for $α_1 α_2 = 1/2$, this representation is unique with the exception of a countable set of points. For the last case, we find the basic metric relation and describe the metric properties of a set of numbers whose $A_2$-continued fraction representation does not contain a given combination of two elements. The properties of a random variable for which the elements of its $A_2$-continued fraction representation form a homogeneous Markov chain are also investigated.

### 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.

### Singular and fractal properties of distributions of random variables digits of polybasic representations of which a form homogeneous Markov chain

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 368-374

We study the fractal properties of distributions of random variables digits of polybasic *Q*-representations (*a* generalization of *n*-adic digits) of which form a homogeneous Markov chain in the case where the matrix of transition probabilities contains at least one zero.

### On finite convolutions of singular distributions and a “singular analog” of the Jessen-Wintner theorem

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1082–1088

We study the fractal properties of a convolution of two Cantor distributions. By using the method of characteristic functions, we establish sufficient conditions for the singularity of the convolution of an arbitrary finite number of distributions of random variables with independent *s*-adic digits. We disprove the hypothesis on the validity of a “singular analog” of the Jessen-Wintner theorem for anomalously fractal distributions.

### One class of singular complex-valued random variables of the Jessen-Wintner type

O. V. Shkol’nyi, Pratsiovytyi M. V.

Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1653–1660

We study the structure of the distribution of a complex-valued random variable ξ = Σ*a* _{ k } ξ_{ k }, where ξ_{ k } are independent complex-valued random variables with discrete distribution and *a* _{k} are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity of the distribution of ξ and investigate the fractal properties of the spectrum.

### Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction

Ukr. Mat. Zh. - 1996. - 48, № 8. - pp. 1086-1095

The structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied. We prove that the distribution is pure and the absolute continuity is impossible, give a criterion of singularity, and study the properties of the spectrum. For the distribution of a random variable for which elements of the corresponding continued fraction form a uniform Markov chain, we describe the spectrum, obtain formulas for the distribution function and density, give a criterion of the Cantor property, and prove that an absolutely continuous component is absent.

### 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.