Ланцюгові $A_3$-дроби: основи метричної теорії
We study a geometry of representation of numbers in terms of continued $A_3$-fractions, that is continued fractions, elements of which are acquire from a set $A_3\equiv\{s_0,s_1,s_2\}$, де $0<s_0<s_1<s_2, s_i\in\mathbb{R}$. We prove that if $s_0s_2=\frac{4}{3}$ and $...
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| Date: | 2017 |
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| Keywords: | keywords |
| Main Authors: | , , |
| Format: | Article |
| Language: | Ukrainian |
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Інститут математики НАН України
2017
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| Online Access: | https://trim.imath.kiev.ua/index.php/trim/article/view/408 |
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| Journal Title: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Transactions of Institute of Mathematics of NAS of Ukraine| Summary: | We study a geometry of representation of numbers in terms of continued $A_3$-fractions, that is continued fractions, elements of which are acquire from a set $A_3\equiv\{s_0,s_1,s_2\}$, де $0<s_0<s_1<s_2, s_i\in\mathbb{R}$. We prove that if $s_0s_2=\frac{4}{3}$ and $s_1=(s_0+s_2)/2$ then each point of a certain interval has no more than two $A_3$-representation, and the set of points having two representations is countable, consequently, the encoding numbers system by means of a three-character alphabet, which is based on a decomposition of numbers in such continued fractions, has zero redundancy. The emphasis in the work is given to the topological-metric aspect of this representation (geometric sense of a figures, properties of the cylindrical and tail sets and so on). |
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