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 $T_s$ of "particularly nonnormal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose $s$-adic digits have the asymptotic frequencies in the nonter...
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| Date: | 2005 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2005
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3676 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | Dedicated to V. S. Korolyuk on occasion of his 80-th birthday
Properties of the set $T_s$ of "particularly nonnormal numbers" of the unit interval are studied in details ($T_s$ 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 $T_s$ 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 ( $T_s$ 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.
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