INFINITE-SYMBOL B-REPRESENTATION OF REAL NUMBERS AND SOME OF ITS APPLICATIONS

Bukovinsʹkij matematičnij žurnal Pub Date : 2023-01-01 DOI:10.31861/bmj2023.01.08

M. Pratsiovytyi, O. Bondarenko, N. Vasylenko, I. Lysenko

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Abstract

In the paper we justify existence and unity $B$-representation of numbers of segment $(0;1)$, which uses as a basis a positive number $a$ that satisfies the condition $00~\forall n\in Z$, $\sum\limits_{n=-\infty}^{+\infty}\Theta_n=1$, $b_{n+1}\equiv\sum\limits_{i=-\infty}^{n-1}=b_n+\Theta_n$ $\forall n\in Z$. The geometry of $B$-representations of numbers is described (geometric content of numbers, properties of cylinder and tail sets, topological and metric properties of sets with restrictions on the use of numbers). The left and right shift operators of numbers are studied, a group of continuous transformations of the unit interval preserving the tails of the $B$-representation of numbers is described.

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实数的无穷符号b表示及其一些应用

在本文中，我们证明了存在和统一 $B$-表示段的数量 $(0;1)$，它使用一个正数作为基 $a$ 它满足这个条件 $0<a<\frac{1}{3}$ 特别是正根 $\tau$ 方程的 $x^2+x-1=0$，双侧序列 $(\Theta_n)$： $\Theta_0=\frac{1-3a}{1-a}$， $\Theta_{-n}=\Theta_n=a^{|n|}$ 还有字母表 $Z=\{0,\pm 1, \pm 2, \pm, \dots \},$\\ 即$$x=b_{\alpha_1}+\sum\limits_{k=2}^{m}b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i}\equiv \Delta^{B}_{\alpha_1\alpha_2...\alpha_m(\emptyset)},$$$$x=b_{\alpha_1}+\sum\limits_{k=2}^{\infty}b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i}\equiv \Delta^{B}_{\alpha_1\alpha_2...\alpha_n...},$$其中$\alpha_n\in Z$$\Theta_n>0~\forall n\in Z$$\sum\limits_{n=-\infty}^{+\infty}\Theta_n=1$$b_{n+1}\equiv\sum\limits_{i=-\infty}^{n-1}=b_n+\Theta_n$$\forall n\in Z$。描述了$B$ -数字表示的几何(数字的几何内容，柱面集和尾集的性质，具有数字使用限制的集合的拓扑和度量性质)。研究了数字的左移算子和右移算子，给出了一组保持数字$B$ -表示尾部的单位区间连续变换。

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Bukovinsʹkij matematičnij žurnal

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