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Keywords:
run-length function; Bolyai-Rényi expansion; Lebesgue measure; Hausdorff dimension
run-length function; Bolyai-Rényi expansion; Lebesgue measure; Hausdorff dimension
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 $.
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