A two-symbol system of encoding and some of its applications
The article is devoted to a two-symbol system of encoding for real numbers with two bases of different signs \(g_0 \in (0;\frac{1}{2}]\) and \(g_1\equiv g_0-1\), as well as its applications in metric number theory and the metric theory of functions. We prove that any natural number \(a\) can be repr...
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| Дата: | 2025 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2025
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2395 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| id |
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admjournalluguniveduua-article-23952025-10-27T20:24:52Z A two-symbol system of encoding and some of its applications Pratsiovytyi, Mykola Lysenko, Iryna Ratushniak, Sofiia two-symbol system of encoding (representation) of numbers, \(G_2\)-representation of numbers, cylindrical set (cylinder), left and right shift operator, tail set, group of continuous transformations of real interval preserving tails 11H71, 26A46, 93B17 The article is devoted to a two-symbol system of encoding for real numbers with two bases of different signs \(g_0 \in (0;\frac{1}{2}]\) and \(g_1\equiv g_0-1\), as well as its applications in metric number theory and the metric theory of functions. We prove that any natural number \(a\) can be represented as $$a=2^n+\sum\limits_{k=1}^{n}[(-1)^{1+\sigma_k}a_k2^{n-k}]\equiv (1a_1\ldots a_n)_{G},$$ where \(a_k\in \{0;1\}\) and \(\sigma_k=a_1+\ldots+a_{k-1}\), and there exist exactly two such representations. Any number \(x\in (0;g_0]\) can be represented as$$\delta_{\alpha_1}+\sum\limits_{k=2}^{\infty}(\delta_{\alpha_k}\prod\limits_{j=1}^{k-1}g_{\alpha_j})\equiv\Delta^{G_2}_{\alpha_1\alpha_2\ldots\alpha_k\ldots}, \delta_{\alpha_k}=\alpha_kg_{1-\alpha_k}.$$ Most numbers have a unique \(G_2\)-representation, while a countable set has exactly two representations: \(\Delta^{G_2}_{c_1\ldots c_m01(0)}=\Delta^{G_2}_{c_1\ldots c_m11(0)}\). For \(g_0=\frac12\), any number \(x\) in the interval \([0;1]\) has the expansion$$x=\frac{1}{2}\alpha_0+\sum\limits_{k=1}^{\infty}\frac{\alpha_k(-1)^{1+\sigma_k}}{2^k}\equiv\Delta_{\alpha_0\alpha_1\ldots\alpha_n\ldots}.$$ Lugansk National Taras Shevchenko University 2025-10-27 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2395 10.12958/adm2395 Algebra and Discrete Mathematics; Vol 40, No 1 (2025) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2395/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2395/1300 Copyright (c) 2025 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2025-10-27T20:24:52Z |
| collection |
OJS |
| language |
English |
| topic |
two-symbol system of encoding (representation) of numbers \(G_2\)-representation of numbers cylindrical set (cylinder) left and right shift operator tail set group of continuous transformations of real interval preserving tails 11H71 26A46 93B17 |
| spellingShingle |
two-symbol system of encoding (representation) of numbers \(G_2\)-representation of numbers cylindrical set (cylinder) left and right shift operator tail set group of continuous transformations of real interval preserving tails 11H71 26A46 93B17 Pratsiovytyi, Mykola Lysenko, Iryna Ratushniak, Sofiia A two-symbol system of encoding and some of its applications |
| topic_facet |
two-symbol system of encoding (representation) of numbers \(G_2\)-representation of numbers cylindrical set (cylinder) left and right shift operator tail set group of continuous transformations of real interval preserving tails 11H71 26A46 93B17 |
| format |
Article |
| author |
Pratsiovytyi, Mykola Lysenko, Iryna Ratushniak, Sofiia |
| author_facet |
Pratsiovytyi, Mykola Lysenko, Iryna Ratushniak, Sofiia |
| author_sort |
Pratsiovytyi, Mykola |
| title |
A two-symbol system of encoding and some of its applications |
| title_short |
A two-symbol system of encoding and some of its applications |
| title_full |
A two-symbol system of encoding and some of its applications |
| title_fullStr |
A two-symbol system of encoding and some of its applications |
| title_full_unstemmed |
A two-symbol system of encoding and some of its applications |
| title_sort |
two-symbol system of encoding and some of its applications |
| description |
The article is devoted to a two-symbol system of encoding for real numbers with two bases of different signs \(g_0 \in (0;\frac{1}{2}]\) and \(g_1\equiv g_0-1\), as well as its applications in metric number theory and the metric theory of functions. We prove that any natural number \(a\) can be represented as $$a=2^n+\sum\limits_{k=1}^{n}[(-1)^{1+\sigma_k}a_k2^{n-k}]\equiv (1a_1\ldots a_n)_{G},$$ where \(a_k\in \{0;1\}\) and \(\sigma_k=a_1+\ldots+a_{k-1}\), and there exist exactly two such representations. Any number \(x\in (0;g_0]\) can be represented as$$\delta_{\alpha_1}+\sum\limits_{k=2}^{\infty}(\delta_{\alpha_k}\prod\limits_{j=1}^{k-1}g_{\alpha_j})\equiv\Delta^{G_2}_{\alpha_1\alpha_2\ldots\alpha_k\ldots}, \delta_{\alpha_k}=\alpha_kg_{1-\alpha_k}.$$ Most numbers have a unique \(G_2\)-representation, while a countable set has exactly two representations: \(\Delta^{G_2}_{c_1\ldots c_m01(0)}=\Delta^{G_2}_{c_1\ldots c_m11(0)}\). For \(g_0=\frac12\), any number \(x\) in the interval \([0;1]\) has the expansion$$x=\frac{1}{2}\alpha_0+\sum\limits_{k=1}^{\infty}\frac{\alpha_k(-1)^{1+\sigma_k}}{2^k}\equiv\Delta_{\alpha_0\alpha_1\ldots\alpha_n\ldots}.$$ |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2025 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2395 |
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AT pratsiovytyimykola atwosymbolsystemofencodingandsomeofitsapplications AT lysenkoiryna atwosymbolsystemofencodingandsomeofitsapplications AT ratushniaksofiia atwosymbolsystemofencodingandsomeofitsapplications AT pratsiovytyimykola twosymbolsystemofencodingandsomeofitsapplications AT lysenkoiryna twosymbolsystemofencodingandsomeofitsapplications AT ratushniaksofiia twosymbolsystemofencodingandsomeofitsapplications |
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2025-12-02T15:42:42Z |
| last_indexed |
2025-12-02T15:42:42Z |
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