Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits
Dedicated to V. S. Korolyuk on occasion of his 80-th birthday Properties of the set Tₛ of "particularly nonnormal numbers" of the unit interval are studied in details (Tₛ consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic e...
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| Дата: | 2005 |
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| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2005
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| Назва видання: | Український математичний журнал |
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| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/165826 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits / M.V. Pratsiovytyi, H.M. Torbin // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1163–1170. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1658262020-02-17T01:27:47Z Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits Pratsiovytyi, M.V. Torbin, H.M. Статті Dedicated to V. S. Korolyuk on occasion of his 80-th birthday Properties of the set Tₛ of "particularly nonnormal numbers" of the unit interval are studied in details (Tₛ consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proven that the set Tₛ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ( Tₛ is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented. Детально вивчаються властивості множини Tₛ „особливо ненормальних чисел" одиничного інтервалу (тобто множини чисел x, для яких немає асимптотичної частоти деяких цифр в s-адичному зображенні, а деякі цифри мають асимптотичні частоти). Доведено, що множина Tₛ є нехтуваною в топологічному сенсі (першої категорії Бера) та загальною в сенсі фрактальної геометрії (Tₛ є суперфрактальною множиною, розмірність Хаусдорфа-Безиковича якої дорівнює одиниці). Наведено топологічну і фрактальну класифікацію множин дійсних чисел через аналіз асимптотичної частоти їх s-адичних зображень. 2005 Article Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits / M.V. Pratsiovytyi, H.M. Torbin // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1163–1170. — Бібліогр.: 8 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165826 519.21 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Pratsiovytyi, M.V. Torbin, H.M. Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits Український математичний журнал |
| description |
Dedicated to V. S. Korolyuk on occasion of his 80-th birthday
Properties of the set Tₛ of "particularly nonnormal numbers" of the unit interval are studied in details (Tₛ consists of real numbers x, some of whose s-adic digits have the asymptotic frequencies in the nonterminating s-adic expansion of x, and some do not). It is proven that the set Tₛ is residual in the topological sense (i.e., it is of the first Baire category) and it is generic in the sense of fractal geometry ( Tₛ is a superfractal set, i.e., its Hausdorff - Besicovitch dimension is equal to 1). A topological and fractal classification of sets of real numbers via analysis of asymptotic frequencies of digits in their s-adic expansions is presented. |
| format |
Article |
| author |
Pratsiovytyi, M.V. Torbin, H.M. |
| author_facet |
Pratsiovytyi, M.V. Torbin, H.M. |
| author_sort |
Pratsiovytyi, M.V. |
| title |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_short |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_full |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_fullStr |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_full_unstemmed |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits |
| title_sort |
singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their s-adic digits |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/165826 |
| citation_txt |
Singular Probability Distributions and Fractal Properties of Sets of Real Numbers Defined by the Asymptotic Frequencies of Their s-Adic Digits / M.V. Pratsiovytyi, H.M. Torbin // Український математичний журнал. — 2005. — Т. 57, № 9. — С. 1163–1170. — Бібліогр.: 8 назв. — англ. |
| series |
Український математичний журнал |
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2023-10-18T22:17:04Z |
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2023-10-18T22:17:04Z |
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