Run-length function of the Bolyai-Rényi expansion of real numbers

@article{Li2024,
abstract = {By iterating the Bolyai-Rényi transformation $T(x)=(x+1)^\{2\} \hspace\{4.44443pt\}(\@mod \; 1)$, 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 \lbrace 0,1,2\rbrace $ for all $n\in \mathbb \{N\}$. For any real number $x\in [0,1)$ and digit $i\in \lbrace 0,1,2\rbrace $, 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 \lbrace 0,1,2\rbrace $, the Lebesgue measure of the set \[ D(i)=\Bigl \lbrace x\in [0,1)\colon \lim \_\{n\rightarrow \infty \} \frac\{r\_n(x,i)\}\{\log n\}=\frac\{1\}\{\log \theta \_\{i\}\} \Bigr \rbrace \] is $1$, where $\theta _\{i\}=1+\sqrt\{4i+1\}$. We also obtain that the level set \[ E\_\{\alpha \}(i)=\Bigl \lbrace x\in [0,1)\colon \lim \_\{n\rightarrow \infty \} \frac\{r\_n(x,i)\}\{\log n\}=\alpha \Bigr \rbrace \] is of full Hausdorff dimension for any $0\le \alpha \le \infty $.},
author = {Li, Rao, Lü, Fan, Zhou, Li},
journal = {Czechoslovak Mathematical Journal},
keywords = {run-length function; Bolyai-Rényi expansion; Lebesgue measure; Hausdorff dimension},
language = {eng},
number = {1},
pages = {319-335},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Run-length function of the Bolyai-Rényi expansion of real numbers},
url = {http://eudml.org/doc/299246},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Li, Rao
AU - Lü, Fan
AU - Zhou, Li
TI - Run-length function of the Bolyai-Rényi expansion of real numbers
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 1
SP - 319
EP - 335
AB - By iterating the Bolyai-Rényi transformation $T(x)=(x+1)^{2} \hspace{4.44443pt}(\@mod \; 1)$, 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 \lbrace 0,1,2\rbrace $ for all $n\in \mathbb {N}$. For any real number $x\in [0,1)$ and digit $i\in \lbrace 0,1,2\rbrace $, 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 \lbrace 0,1,2\rbrace $, the Lebesgue measure of the set \[ D(i)=\Bigl \lbrace x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac{r_n(x,i)}{\log n}=\frac{1}{\log \theta _{i}} \Bigr \rbrace \] is $1$, where $\theta _{i}=1+\sqrt{4i+1}$. We also obtain that the level set \[ E_{\alpha }(i)=\Bigl \lbrace x\in [0,1)\colon \lim _{n\rightarrow \infty } \frac{r_n(x,i)}{\log n}=\alpha \Bigr \rbrace \] is of full Hausdorff dimension for any $0\le \alpha \le \infty $.
LA - eng
KW - run-length function; Bolyai-Rényi expansion; Lebesgue measure; Hausdorff dimension
UR - http://eudml.org/doc/299246
ER -