HAUSDORFF DIMENSION OF THE SET CONCERNING WITH BOREL-BERNSTEIN THEORY IN L¨UROTH EXPANSIONS
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초록
It is well known that every $x\in(0,1]$ can be expanded to an infinite L\"{u}roth series with the form of \[ x=\tfrac{1}{d_1(x)}+\cdots+\tfrac{1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}+\cdots, \] 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\"{u}roth series expansions are considered. Namely, the Hausdorff dimension of following the set $$ F_{\phi}=\{x\in (0,1]: d_n(x)\geq \phi(n), \ {\rm{i. \ o.}} \ n\} $$ is determined, where $\phi$ is an integer-valued function defined on $\mathbb{N}$, and $\phi(n)\to \infty$ as $n\to \infty$.
Abstract
It is well known that every $x\in(0,1]$ can be expanded to an infinite L\"{u}roth series with the form of \[ x=\tfrac{1}{d_1(x)}+\cdots+\tfrac{1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}+\cdots, \] 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\"{u}roth series expansions are considered. Namely, the Hausdorff dimension of following the set $$ F_{\phi}=\{x\in (0,1]: d_n(x)\geq \phi(n), \ {\rm{i. \ o.}} \ n\} $$ is determined, where $\phi$ is an integer-valued function defined on $\mathbb{N}$, and $\phi(n)\to \infty$ as $n\to \infty$.
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