Article
| Title: | Hausdorff dimension of the maximal run-length in dyadic expansion (English) |
| Author: | Zou, Ruibiao |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 |
| Volume: | 61 |
| Issue: | 4 |
| Year: | 2011 |
| Pages: | 881-888 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | For any $x\in [0,1)$, let $x=[\epsilon _1,\epsilon _2,\cdots ,]$ be its dyadic expansion. Call $r_n(x):=\max \{j\geq 1\colon \epsilon _{i+1}=\cdots =\epsilon _{i+j}=1$, $0\leq i\leq n-j\}$ the $n$-th maximal run-length function of $x$. P. Erdös and A. Rényi showed that $\lim _{n\to \infty }{r_n(x)}/{\log _2 n}=1$ almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than $\log _2 n$, is quantified by their Hausdorff dimension. |
| Keyword: | run-length function |
| Keyword: | Hausdorff dimension |
| Keyword: | dyadic expansion |
| MSC: | 11K55 |
| MSC: | 28A78 |
| MSC: | 28A80 |
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| Date available: | 2011-12-16T15:33:47Z |
| Last updated: | 2012-05-31 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/141793 |
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