| Title:
|
Hausdorff dimension of the maximal run-length in dyadic expansion (English) |
| Author:
|
Zou, Ruibiao |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
61 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
881-888 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
run-length function |
| Keyword:
|
Hausdorff dimension |
| Keyword:
|
dyadic expansion |
| MSC:
|
11K55 |
| MSC:
|
28A78 |
| MSC:
|
28A80 |
| idZBL:
|
Zbl 1249.11085 |
| idMR:
|
MR2886243 |
| DOI:
|
10.1007/s10587-011-0055-5 |
| . |
| Date available:
|
2011-12-16T15:33:47Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141793 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| . |
