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M. V. Pratsiovytyi

Publications and source records attributed to M. V. Pratsiovytyi.

9 recordsLinked to original sources
arXiv2026

An analogue of the Gauss-Kuzmin problem for continued A2-fractions

In this paper, a special chain representation of real numbers on a fixed interval is considered, where the elements of the expansion can take only one of two possible values. For this encoding system, a problem in metric number theory and dynamical systems is solved, which is a direct analogue of the classical Gauss problem for simple continued fractions. Specifically, the asymptotic behavior of the Lebesgue measure for a special class of sets is investigated. These sets are formed by those numbers for which the remainder (or infinite tail) of their chain expansion, after discarding the first few elements, is strictly less than a predetermined value.

math.DS↗
arXiv2026

On one class of nowhere non-monotonic functions with fractal properties that contains a subclass of singular functions

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=δ_{α_1(x)1}+\sum^{\infty}_{k=2}\left[δ_{α_k(x)k}\prod^{k-1}_{j=1}g_{α_j (x)j}\right]\equivΔ^{G^*_3}_{α_1α_2\ldotsα_k\ldots}, $$ where $||q^*_{ik}||$ is given infinite stochastic positive matrix ($i=0,1,2$; $k \in N$); $β_{0k}=0$, $β_{1k}=q_{0k}$, $β_{2k}=q_{0k}+q_{1k}$; $(\varepsilon_k)$ is given sequence of numbers such that $0\leqslant \varepsilon_k \leqslant 1 $; $g_{0k}=\dfrac{1+\varepsilon_k}{3}=g_{2k}$, $g_ {1k}=\dfrac{1-2\varepsilon_k}{3}$, $δ_{0k}=0$, $δ_{1k}=g_{0k}$, $δ_{2k}=g_{0k}+g_{1k}$, $k\in N$. We found criteria of strict monotonicity, non monotonicity and nowhere monotonicity, non-differentiability and singularity of the functions. We pay attention to properties of level sets of the functions.

math.CA↗
arXiv2026

Asymptotic mean of digits of the $Q_s$-representation of the fractional part of a real number and related problems of fractal geometry and fractal analysis

We introduce a concept of asymptotic mean of digits (symbols) in the $Q_s$-representation of a real number, that is a generalization of the $s$-adic representation and have a self-similar geometry. We discuss its relationship with the frequencies of digits and formulate problems related to the concept. We study the topological, metric, and fractal properties of the set of real numbers that have no asymptotic mean of $Q_s$-symbols. Also we study topological, metric and fractal properties of the sets of real numbers that have asymptotic mean of $Q_3$-symbols which is equal to value of digit frequency of number.

math.NT↗
arXiv2026

Linear fractals of the Besicovitch-Eggleston type

We study topological, metric and fractal properties of set of numbers $[0;1]$ with given asymptotic mean of digits in their ternary representation. We investigate connection of these numbers and numbers with a given frequency of digits.

math.NT↗
arXiv2026

Topological, metric and fractal properties of the set of real numbers with a given asymptotic mean of digits in their $4$-adic representation in the case when the digit frequencies exist

In the paper we describe some properties of function $$ y=r(x)=\lim_{n\to\infty}\frac{1}{n}\sum^{\infty}_{k=1}α_k(x), \text{ where } x=\sum^{\infty}_{k=1}α_k(x)4^{-k} $$ of $4$-adic digits asymptotic mean of fractional part of real number $x$, particularly properties of it's level sets $ S_θ=\left\{x: r(x)=θ,\: θ=const, \: 0\leqslantθ\leqslant 3\right\}, $ if all $4$-adic digits frequencies exist, i.e. $$ ν_i(x)=\lim_{n\to\infty}n^{-1}\#\{k: α_k(x)=i, i\leqslant n\}, \:\: i=0,1,2,3. $$ We provided an algorithm of constructing point from the set $S_θ$, and proved continuality and every where density of the set. We found conditions of zero and full Lebesgue measure and estimates of Hausdorff-Besicovitch fractal dimension.

math.NT↗
arXiv2026

Fractal functions defined in terms of number representations in systems with a redundant alphabet

For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of numbers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion $x=\sum\limits_{n=1}^{\infty}s^{-n}α_n=Δ^{r_s}_{α_1α_2...α_n...}.$ The algorithm for expanding a number into such a series is justified in the paper. The geometry of this representation is studied, including the geometric meaning of digits, properties of cylinder sets -- particularly the specificity of their overlaps -- and metric relations, as well as the connection between the representation and partial sums of the corresponding series. The paper also presents results on the study of a function $f$ defined by $f(x=\sum\limits_{n=1}^{\infty}\frac{α_n}{(r+1)^n})=Δ^{r_s}_{α_1α_2...α_n...}, α_n\in A.$ It is proved that the function $f$ is continuous at every point that has a unique representation in the classical numeration system with base $r+1$, and discontinuous at points having two representations. The function has unbounded variation and a self-affine graph. For $r<2s-1$, the function possesses singleton, finite, countable, and continuum level sets, including fractal ones; for $r>2s-2$, every level set is a continuum, and moreover it is fractal or anomalously fractal.

math.NT↗
arXiv2016

Level sets of asymptotic mean of digits function for 4-adic representation of real number

We study topological, metric and fractal properties of the level sets $$S_θ=\{x:r(x)=θ\}$$ of the function $r$ of asymptotic mean of digits of a number $x\in[0;1]$ in its $4$-adic representation, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}α_i(x)$$ if the asymptotic frequency $ν_j(x)$ of at least one digit does not exist, were $$ ν_j(x)=\lim_{n\to\infty}n^{-1}#\{k: α_k(x)=j, k\leqslant n\}, \:\: j=0,1,2,3. $$

math.NT↗