Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers
We describe the geometry of representation of numbers belonging to \((0,1]\) by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semicylinders and tail sets, metric relation...
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| Дата: | 2018 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/716 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-7162018-04-04T09:58:22Z Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers Zhykharyeva, Yulia Pratsiovytyi, Mykola Luroth series, \(L\)-representation, cylinder, semicylinder, shift operator, random variable defined by \(L\)-representation, fractal,Hausdorff-Besicovitch dimension 11K55 We describe the geometry of representation of numbers belonging to \((0,1]\) by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semicylinders and tail sets, metric relations; prove topological, metric and fractal properties of sets of numbers with restrictions on use of ``digits''; show that for determination of Hausdorff-Besicovitch dimension of Borel set it is enough to use connected unions of cylindrical sets of the same rank. Some applications of \(L\)-representation to probabilistic theory of numbers are also considered. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/716 Algebra and Discrete Mathematics; Vol 14, No 1 (2012) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/716/248 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
Luroth series \(L\)-representation cylinder semicylinder shift operator random variable defined by \(L\)-representation fractal,Hausdorff-Besicovitch dimension 11K55 |
| spellingShingle |
Luroth series \(L\)-representation cylinder semicylinder shift operator random variable defined by \(L\)-representation fractal,Hausdorff-Besicovitch dimension 11K55 Zhykharyeva, Yulia Pratsiovytyi, Mykola Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers |
| topic_facet |
Luroth series \(L\)-representation cylinder semicylinder shift operator random variable defined by \(L\)-representation fractal,Hausdorff-Besicovitch dimension 11K55 |
| format |
Article |
| author |
Zhykharyeva, Yulia Pratsiovytyi, Mykola |
| author_facet |
Zhykharyeva, Yulia Pratsiovytyi, Mykola |
| author_sort |
Zhykharyeva, Yulia |
| title |
Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers |
| title_short |
Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers |
| title_full |
Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers |
| title_fullStr |
Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers |
| title_full_unstemmed |
Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers |
| title_sort |
expansions of numbers in positive lüroth series and their applications to metric, probabilistic and fractal theories of numbers |
| description |
We describe the geometry of representation of numbers belonging to \((0,1]\) by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semicylinders and tail sets, metric relations; prove topological, metric and fractal properties of sets of numbers with restrictions on use of ``digits''; show that for determination of Hausdorff-Besicovitch dimension of Borel set it is enough to use connected unions of cylindrical sets of the same rank. Some applications of \(L\)-representation to probabilistic theory of numbers are also considered. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/716 |
| work_keys_str_mv |
AT zhykharyevayulia expansionsofnumbersinpositivelurothseriesandtheirapplicationstometricprobabilisticandfractaltheoriesofnumbers AT pratsiovytyimykola expansionsofnumbersinpositivelurothseriesandtheirapplicationstometricprobabilisticandfractaltheoriesofnumbers |
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2024-04-12T06:26:35Z |
| last_indexed |
2024-04-12T06:26:35Z |
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1796109222057869312 |