arXiv subjects

Mykola Pratsiovytyi

Publications and source records attributed to Mykola Pratsiovytyi.

11 recordsLinked to original sources

arXiv2026

A series with a Cantorval set of subsums and ratio of consecutive terms tending to $1/2$ (the second Jones problem)

The second Jones problem (2011) consists in determining whether there exists a convergent positive series for which the ratio of consecutive terms has a limit and the set of subsums is a Cantorval; in the survey by Gł{\k a}b and Prus-Wiśniowski (2026) this problem is included in the list of open problems. In this paper, a family of such series is constructed explicitly as combinations of blocks of the form $\frac{2^k-1}{2^t} + \frac{2^{k-1}}{2^t} + \frac{2^{k-2}}{2^t} + \cdots + \frac{2}{2^t}$, where $t$ is the index of the last term of the block in the series and $k\ge2$ is the length of the block. For each series of the family it is proved that its set of subsums contains an interval and is not a finite union of closed intervals, and hence is a Cantorval. The proof relies on mixed-radix numerical systems with a redundant alphabet and uses an idea of Guthrie and Nymann (1988). If the lengths of the blocks tend to infinity, then the limit of the ratio of consecutive terms of such series is $\tfrac{1}{2}$, which, by a result of Jones, is the only possible value of the limit for which the set of subsums of a series can be a Cantorval. Thus, an affirmative answer to the second Jones problem is obtained.

math.CA↗

arXiv2026

Infinite continued fractions with a finite alphabet and their applications

The paper investigates the topological and metric properties of the set $E$ of values of all infinite continued fractions of the form \[0+\frac{1}{a_1+\dfrac{1}{a_2+_{\ddots}}}=1/a_1+1/a_2+...+1/a_n+...\equiv[0;a_1,a_2,...,a_n,...],\] whose partial quotients ($a_n$) take values in a finite set of positive real numbers $\{e_0,e_1,...,e_{s-1}\}$, $e_0<e_1<...<e_{s-1}$. It is established that the set $E$ is bounded, has cardinality continuum, and is perfect. Conditions under which $E$ is an interval, a nowhere dense set, or a set of Lebesgue measure zero are established. Necessary and sufficient conditions are obtained for $E$ to be an interval and for the corresponding coding system by such continued fractions to have zero redundancy, i.e., for each number to have at most two representations. The geometric meaning of the digits in this representation, as well as the metric relations, is described in terms of the properties of cylindrical sets (cylinders and cylindrical intervals). It is proved that the basic metric ratio, defined as the ratio of the diameter of a cylinder to the diameter of its parent cylinder, is bounded away from both zero and one.

math.NT↗

arXiv2026

Infinite Bernoulli convolutions generated by multigeometric series and their properties

The paper is devoted to infinite Bernoulli convolutions generated by positive multigeometric series and to probability distributions of random variables whose digits in an even integer base-$s$ expansion with two redundant digits form a sequence of independent and identically distributed random variables. The main objects of the article are random variables: $ξ=\sum\limits_{n=1}^{\infty}\frac{ξ_n}{s^n}$, where $(ξ_n)$ is a sequence of independent and identically distributed random variables taking values $0, 1, 2, \dots, s-1, s, s+1$ with probabilities $p_0$, $p_1$, $p_2, \dots, p_{s-1}, p_s, p_{s+1}$ respectively $(3 0$ and $q_1=1-q_0>0$. We study conditions under which the above random variables have absolutely continuous or singular distributions as well as topological, metric, and fractal properties of their supports. The main focus is on the case where the spectrum is a Cantorval.

math.PR↗

arXiv2026

A class of Tribin functions related to $s$-symbol encodings of numbers with a zero redundancy

In this paper, we consider a continuum class of continuous nowhere monotonic functions that generalize certain non-differentiable functions, including the Bush function, Wunderlich function, continuous Cantor projectors, Tribin function, etc. We consider a construction of the function related to $s$-symbol representations of numbers with a zero redundancy that are topologically equivalent to the classical $s$-adic representation (a value of the function has a two-symbol representation). Moreover, the condition on the first digit of a representation for the value of the function is more general than conditions considered before. The main object of study is a continuous function defined by equality \begin{gather*} f(Δ^{s^*}_{α_1α_2\ldotsα_n\ldots}) = Δ^{2^*}_{β_1β_2\ldotsβ_n\ldots}, \quad α_n \in \{ 0, 1, 2, \ldots, s - 1 \} \equiv A_s, β_1 = \begin{cases} 0 & \text{if $α_1 \in A_0$}, 1 & \text{if $α_1 \in A_1$}, \end{cases} \quad β_{n+1} = \begin{cases} β_n & \text{if $α_{n+1} = α_n$}, 1 - β_n & \text{if $α_{n+1} \neq α_n$}. \end{cases} \end{gather*} where $Δ^{s^*}_{α_1α_2\ldotsα_n\ldots}$ is an $s$-symbol representation of a number $x \in [0, 1]$ that is topologically equivalent to the classical $s$-adic representation, $Δ^{2^*}_{β_1β_2\ldotsβ_n\ldots}$ is a two-symbol representation that is topologically equivalent to the classical binary representation, and $A_0 \cup A_1 = A_s$, $A_0 \neq A_s \neq A_1$.

math.FA↗

arXiv2026

Sets of distinct representations of numbers in numeral systems with a natural base and a redundant alphabet

In this work, we study a numeral system with a natural base $s \geq 2$ and a redundant alphabet $A_r=\{0,1, \dots, r\}$, where $s \leq r \leq 2s-2$. We investigate the topological, metric, and fractal properties of the set of numbers in the interval $\left[0,\frac{r}{s-1}\right]$ that admit a unique representation $x=\sum\limits_{n=1}^{\infty}\frac{α_n} {s^n}\equivΔ^{r_s}_{α_1α_2...α_n...}$, $α_n\in A_r$. The criterion for the uniqueness of the number representation is established. It is proved that the Hausdorff--Besicovitch dimension of the set of numbers with a unique representation is equal to $\frac{\ln(2s-r-1)}{\ln s}$. An analysis of the quantity of representations of numbers having purely periodic representations with a simple period (a single-digit period) is carried out. It is proved that the set of numbers that admit a continuum of distinct representations has full Lebesgue measure. Conditions for a number to belong to this set are given in terms of one of its representations.

math.NT↗

arXiv2026

Singular distributions of random variables with independent digits of representation in numeral system with natural base and redundant alphabet

Given natural parameters s and r, where $2\leq s\leq r$, we consider the distribution of a random variable $ξ=\sum\limits_{k=1}^{\infty}s^{-k}ξ_k\equivΔ^{r_s}_{ξ_1ξ_2...ξ_k...},$ where $(ξ_k)$ is a sequence of independent random variables taking values in $\{0,1,...,r\}$ with probabilities $p_0,p_1,...,p_r$, respectively, and all $ p_i<1$. In the case s=3=r, necessary and sufficient conditions for the singularity and absolute continuity of the distribution of random variable are established. The work also discusses the connection between the distribution of random variable and infinite Bernoulli convolutions governed by the corresponding series as well as representations of numbers in the base-3 numeral system with one redundant digit. Several open problems are formulated.

math.PR↗

arXiv2024

Fractal analysis of Guthrie-Nymann's set and its generalisations

In this paper, we study the fractal properties of the boundary of the Cantorval connected with Guthrie-Nymann's series. In particular, we prove that such a Cantorval can be represented as a union of open intervals and a Cantor set having zero Lebesgue measure and a fractional Hausdorff dimension. Moreover, we extend the result to a countable family of Cantorvals with a similar structure.

math.DS↗

arXiv2023

Cantorvals as sets of subsums for a series connected with trigonometric functions

We study properties of the set of subsums for a convergent series $ k_1 \sin x + \dots + k_m \sin x +\dots + k_1\sin x^n +\dots + k_m \sin x^n + \dots $, where $k_1, k_2, k_3,\dots,k_m$ are fixed positive integers and $0<x<1$. Depends on parameter $x$ this set can be a finite union of closed intervals or Cantor-type set or even Cantorval.

math.NT↗

arXiv2020

One class of continuous functions related to Engel series and having complicated local properties

We construct and study the class of continuous on $[0, 1]$ functions with continuum set of peculiarities (singular, nowhere monotonic, and non-differentiable functions are among them). The representative of this class is the function defined by the Engel representation of argument and some convergent real series. We study local and global properties of this function: structural, extremal, differential, integral, and fractal properties.

math.FA↗

arXiv2015

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

We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} ξ= \sum_{k=1}^\infty \frac{(-1)^{k+1}ξ_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 $\{ξ_k\}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $ξ$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_ξ)=0$) 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 $μ_ξ$ are studied in details. Conditions for the Hausdorff--Billingsley dimension preservation on the spectrum by its probability distribution function are also obtained.

math.PR↗

arXiv2007

$\widetilde{Q}$-representation of real numbers and fractal probability distributions

A $\widetilde{Q}-$representation of real numbers is introduced as a generalization of the $p-$adic and $Q-$representations. It is shown that the $\widetilde{Q}-$representation may be used as a convenient tool for the construction and study of fractals and sets with complicated local structure. Distributions of random variables $ξ$ with independent $\widetilde{Q}-$symbols are studied in details. Necessary and sufficient conditions for the probability measures $μ_ξ$ associated with $ξ$ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the $\widetilde{Q}-$representation. In addition the metric-topological properties for the distribution of $ξ$ are investigated. A number of examples are presented.

math.PR↗