Some Sets of GCF$_\epsilon$ Expansions Whose Parameter $\epsilon$ Fetch the Marginal Value

Abstract:

      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.