Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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HAUSDORFF DIMENSION OF THE SET CONCERNING WITH BOREL-BERNSTEIN THEORY IN LÜROTH EXPANSIONS

  • Shen, Luming (Science College Hunan Agricultural University)
  • Received : 2016.07.26
  • Published : 2017.07.01
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Abstract

It is well known that every $x{\in}(0,1]$ can be expanded to an infinite $L{\ddot{u}}roth$ series with 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, the set of points with some restrictions on the digits in $L{\ddot{u}}roth$ series expansions are considered. Namely, the Hausdorff dimension of following the set $$F_{\phi}=\{x{\in}(0,1]\;:\;d_n(x){\geq}{\phi}(n),\;i.o.n}$$ is determined, where ${\phi}$ is an integer-valued function defined on ${\mathbb{N}}$, and ${\phi}(n){\rightarrow}{\infty}$ as $n{\rightarrow}{\infty}$.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China, Hunan Agricultural University, Natural Science Foundation of Hunan Province

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