Some Sets of GCF$_\epsilon$ Expansions Whose Parameter $\epsilon$ Fetch the Marginal Value
Received:July 24, 2014  Revised:December 22, 2014
Key Words: $GCF_\epsilon$ expansions   metric properties   Hausdorff dimension
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11361025).
 Author Name Affiliation Liang TANG Department of Mathematics, Jishou University, Hunan 427000, P. R. China Peijuan ZHOU Department of Mathematics, Jishou University, Hunan 427000, P. R. China Ting ZHONG Department of Mathematics, Jishou University, Hunan 427000, P. R. China
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Let $\epsilon: \mathbb{N}\to \mathbb{R}$ be a parameter function satisfying the condition $\epsilon(k)+k+1> 0$ and let $T_{\epsilon}:(0,1]\to (0,1]$ be a transformation defined by $$T_{\epsilon}(x)=\frac{-1+(k+1)x}{1+k-k\epsilon x} \ {\text{for}}\ x\in \Big(\frac{1}{k+1},\frac{1}{k}\Big].$$ Under the algorithm $T_{\epsilon}$, every $x\in (0,1]$ is attached an expansion, called generalized continued fraction (GCF$_{\epsilon}$) expansion with parameters by Schweiger. Define the sequence $\{k_n(x)\}_{n\ge 1}$ of the partial quotients of $x$ by $k_1(x)=\lfloor 1/x\rfloor$ and $k_n(x)=k_1(T_{\epsilon}^{n-1}(x))$ for every $n\ge 2$. Under the restriction $-k-1<\epsilon(k)<-k$, define the set of non-recurring GCF$_{\epsilon}$ expansions as $$\mathcal{F}_{\epsilon}=\{x\in (0,1]: k_{n+1}(x)>k_n(x)\ {\text{for infinitely many }}\ n\}.$$ It has been proved by Schweiger that $\mathcal{F}_{\epsilon}$ has Lebesgue measure 0. In the present paper, we strengthen this result by showing that \begin{eqnarray*} \left\{\begin{array}{ll}\dim_H \mathcal{F}_{\epsilon}\ge \frac{1}{2}, & \text{when $\epsilon(k)=-k-1+\rho$ for a constant $0<\rho<1$;} \frac{1}{s+2}\le\dim_H \mathcal{F}_{\epsilon}\le \frac{1}{s}, & \text{when $\epsilon(k)=-k-1+\frac{1}{k^s}$ for any $s\ge1$}\end{array}\right.\end{eqnarray*} where $\dim_H$ denotes the Hausdorff dimension.