The fractional dimensional theory in Lüroth expansion

Shen, Luming, and Fang, Kui. "The fractional dimensional theory in Lüroth expansion." Czechoslovak Mathematical Journal 61.3 (2011): 795-807. <http://eudml.org/doc/196558>.

@article{Shen2011,
abstract = {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 $d_n(x)\ge 2$ for all $n\ge 1$. 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 \[ F\_\{\phi \}=\lbrace x\in (0,1]\colon d\_n(x)\ge \phi (n), \ \forall n\ge 1\rbrace \] are completely determined, where $\phi $ is an integer-valued function defined on $\mathbb \{N\}$, and $\phi (n)\rightarrow \infty $ as $n\rightarrow \infty $.},
author = {Shen, Luming, Fang, Kui},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lüroth series; Cantor set; Hausdorff dimension; Lüroth series; Cantor set; Hausdorff dimension},
language = {eng},
number = {3},
pages = {795-807},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The fractional dimensional theory in Lüroth expansion},
url = {http://eudml.org/doc/196558},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Shen, Luming
AU - Fang, Kui
TI - The fractional dimensional theory in Lüroth expansion
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 795
EP - 807
AB - 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 $d_n(x)\ge 2$ for all $n\ge 1$. 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 \[ F_{\phi }=\lbrace x\in (0,1]\colon d_n(x)\ge \phi (n), \ \forall n\ge 1\rbrace \] are completely determined, where $\phi $ is an integer-valued function defined on $\mathbb {N}$, and $\phi (n)\rightarrow \infty $ as $n\rightarrow \infty $.
LA - eng
KW - Lüroth series; Cantor set; Hausdorff dimension; Lüroth series; Cantor set; Hausdorff dimension
UR - http://eudml.org/doc/196558
ER -