arXiv subjects

S. P. Ratushniak

Publications and source records attributed to S. P. Ratushniak.

4 recordsLinked to original sources

arXiv2026

A Continuous digit projector from binary representations of numbers onto $A_2$-representation

As is known, the $A_2$-continued representation of numbers is not topologically equivalent to the classical binary representation; therefore, the digit projector of such representations is a discontinuous function. In this paper, we introduce a continuous function that serves as an analogue of the digit projector of the classical binary representation of numbers into the digits of the $A_2$-continued representation with zero redundancy, namely a function of the form \[f(Δ^2_{α_1α_2...α_{2n-1}α_{2n}...})= Δ^{A_2}_{(\frac{1}{2})^{1-α_1}(\frac{1}{2})^{α_2}... (\frac{1}{2})^{1-α_{2n-1}}(\frac{1}{2})^{α_{2n}}...}, α_n\in \{0,1\}.\] It is proved that the function $f$ is well-defined, continuous, and monotone. Using the normal properties of numbers with respect to their binary representation and Lebesgue's theorem asserting the existence of a finite derivative for a continuous monotone function almost everywhere, the singularity of the function $f$ is established. The paper also establishes a relationship between the considered function, the right-shift operator on digits, and the inversor of the continued representation of numbers. This relationship is then used to establish the singularity of the inversor.

math.FA↗

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

One continuum class of fractal functions defined in terms of $Q^*_s$-representation

In the paper we study a class $F$ of multiparameter functions defined in terms of a polybasic $s$-adic $Q^{*}_{s}$-representation of numbers by \begin{equation*} f_a\bigl(x=Δ^{Q^{*}_s}_{α_1α_2\ldotsα_n\ldots}\bigr) = Δ^{Q^{*}s}_{|a_1-α_1|\,|a_2-α_2|\,\ldots\,|a_n-α_n|\ldots}, \end{equation*} where $(a_n)$ is the sequence of digits for $s$-adic representation of the parameter $a\in[0,1]$, and \begin{equation*} Δ^{Q^{*}_s}_{α_1α_2\ldotsα_n\ldots}= β_{α_1 1}+ \sum_{n=2}^{\infty} \left( β_{α_n n} \prod_{j=1}^{n-1} q_{α_j j} \right) \end{equation*} is the $Q^{*}_{s}$-representation of real numbers generated by a positive stochastic matrix $\|q_{ij}\|$ with $β_{α_n n}=\sum\limits_{i=0}^{α_n-1} q_{in}$. In this paper we investigate the continuity of the function $f_a$ on the sets of $Q^{*}_{s}$-binary and $Q^{*}_{s}$-unary numbers. We prove that the functions in this class are continuous on the set of numbers with a unique $Q^{*}_{s}$-representation. Furthermore, we show that except for $f_0$ and $f_1$, all functions have a countable set of discontinuities at $Q^{*}_{s}$-binary points. We classify the topological types of the value sets of $f_a$ depending on the parameter $a$. We prove that, if the value set is of Cantor type, then it is zero-dimensional. We describe the structural properties of the level sets of $f_a$ in terms of the digits of the $s$-adic representation of $a$. In particular, we establish that a level set of the function $f_a$ can be an empty set, a finite set, or a continuum. For certain values of $s$ we provide examples of fractal level sets and calculate its fractal dimensions.

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↗