Канторвали як множини неелементарних ланцюгових дробів з обмеженим алфавітом
Let $G_{\mathcal{A}}$ be a set of values of continued fractions whose elements belong to a bounded set $\mathcal{A}$ of positive real numbers. We prove that $G_{\mathcal{A}}$ is a continuum bounded and perfect set. For $\mathcal{A}_3=\{0{,}5; 1; 8\}$, the set $G_{\mathcal{A}}$ is a Cantorval, namely...
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| Datum: | 2019 |
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| Ключові слова: | keywords |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Ukrainisch |
| Veröffentlicht: |
Інститут математики НАН України
2019
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| Online Zugang: | https://trim.imath.kiev.ua/index.php/trim/article/view/523 |
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| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Transactions of Institute of Mathematics of NAS of Ukraine| Zusammenfassung: | Let $G_{\mathcal{A}}$ be a set of values of continued fractions whose elements belong to a bounded set $\mathcal{A}$ of positive real numbers. We prove that $G_{\mathcal{A}}$ is a continuum bounded and perfect set. For $\mathcal{A}_3=\{0{,}5; 1; 8\}$, the set $G_{\mathcal{A}}$ is a Cantorval, namely, it is homeomorphic to the set$$E= \{x: x=\sum\limits_{k=1}^{\infty}(\frac{3\alpha_{2k-1}}{4^k}+\frac{2\alpha_{2k}}{4^k}),\alpha_k\in\{0,1\}\},$$where $E$ contains a finite set of intervals whose complements are continuum nowhere dense sets. |
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