Geodesic
The fractional dimensional theory in Lüroth expansion
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 3, pp. 795-807
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It is well known that every $x\in (0,1]$ can be expanded to an infinite Lüroth series in the form of $$x=\frac {1}{d_1(x)}+\cdots +\frac {1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}+\cdots , $$ where $d_n(x)\geq 2$ for all $n\geq 1$. In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $$ F_{\phi }=\{x\in (0,1]\colon d_n(x)\geq \phi (n), \ \forall n\geq 1\} $$are completely determined, where $\phi $ is an integer-valued function defined on $\mathbb N$, and $\phi (n)\to \infty $ as $n\to \infty $.
It is well known that every $x\in (0,1]$ can be expanded to an infinite Lüroth series in the form of $$x=\frac {1}{d_1(x)}+\cdots +\frac {1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}+\cdots , $$ where $d_n(x)\geq 2$ for all $n\geq 1$. In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $$ F_{\phi }=\{x\in (0,1]\colon d_n(x)\geq \phi (n), \ \forall n\geq 1\} $$are completely determined, where $\phi $ is an integer-valued function defined on $\mathbb N$, and $\phi (n)\to \infty $ as $n\to \infty $.
DOI : 10.1007/s10587-011-0028-8
Classification : 11K55, 28A78, 28A80
Keywords: Lüroth series; Cantor set; Hausdorff dimension 
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Shen, Luming; Fang, Kui. The fractional dimensional theory in Lüroth expansion. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 3, pp. 795-807. doi: 10.1007/s10587-011-0028-8

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