| Title:
|
Run-length function of the Bolyai-Rényi expansion of real numbers (English) |
| Author:
|
Li, Rao |
| Author:
|
Lü, Fan |
| Author:
|
Zhou, Li |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
74 |
| Issue:
|
1 |
| Year:
|
2024 |
| Pages:
|
319-335 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
By iterating the Bolyai-Rényi transformation $T(x)=(x+1)^{2} \pmod 1$, almost every real number $x\in [0,1)$ can be expanded as a continued radical expression $$ x=-1+\sqrt {x_{1}+\sqrt {x_{2}+\cdots +\sqrt {x_{n}+\cdots }}} $$ with digits $x_{n}\in \{0,1,2\}$ for all $n\in \mathbb {N}$. For any real number $x\in [0,1)$ and digit $i\in \{0,1,2\}$, let $r_{n}(x,i)$ be the maximal length of consecutive $i$'s in the first $n$ digits of the Bolyai-Rényi expansion of $x$. We study the asymptotic behavior of the run-length function $r_{n}(x,i)$. We prove that for any digit $i\in \{0,1,2\}$, the Lebesgue measure of the set $$ D(i)=\Bigl \{x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac {r_n(x,i)}{\log n}=\frac {1}{\log \theta _{i}} \Bigr \} $$ is $1$, where $\theta _{i}=1+\sqrt {4i+1}$. We also obtain that the level set $$ E_{\alpha }(i)=\Bigl \{x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac {r_n(x,i)}{\log n}=\alpha \Bigr \} $$ is of full Hausdorff dimension for any $0\leq \alpha \leq \infty $. (English) |
| Keyword:
|
run-length function |
| Keyword:
|
Bolyai-Rényi expansion |
| Keyword:
|
Lebesgue measure |
| Keyword:
|
Hausdorff dimension |
| MSC:
|
11K55 |
| MSC:
|
28A80 |
| idZBL:
|
Zbl 07893382 |
| idMR:
|
MR4717837 |
| DOI:
|
10.21136/CMJ.2023.0351-23 |
| . |
| Date available:
|
2024-03-13T10:12:00Z |
| Last updated:
|
2026-04-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152283 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
