arXiv subjects

Sofiia Ratushniak

Publications and source records attributed to Sofiia Ratushniak.

3 recordsLinked to original sources

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

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

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↗