arXiv subjects

O. P. Makarchuk

Publications and source records attributed to O. P. Makarchuk.

4 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

Transformations and functions that preserve the asymptotic mean of digits in the ternary representation of a number

In the paper we study transformations of the interval $[0;1)$ and functions that preserve the asymptotic mean $r$ of the digits in the $s$--adic representation of a number $x$, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}α_i(x)$$ specify the necessary and sufficient conditions for a transformation to belong to this class

math.NT↗

arXiv2026

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

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.

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↗