Journal of the
Korean Mathematical Society JKMS

For generalized continued fraction (GCF) with parameter $\epsilon(k)$, we consider the size of the set whose partial quotients increase rapidly, namely the set $$ E_{\epsilon}(\alpha):=\Big\{x\in (0,1]: k_{n+1}(x)\ge k_n(x)^{\alpha} \ {\text{for all}}\ n\ge 1\Big\}, $$ where $\alpha>1$. We in \cite{ZT} have obtained the Hausdorff dimension of $E_{\epsilon}(\alpha)$ when $\epsilon(k)$ is constant or $\epsilon(k)\sim k^\beta$ for any $\beta\ge1$. As its supplement, now we show that: \begin{align*} \dim_H E_{\epsilon}(\alpha)=\left\{ \begin{array}{ll} \frac{1}{\alpha}, & \hbox{when $-k^\delta\le\epsilon(k)\le k$ with $0\le\delta<1$;} \\ \frac{1}{\alpha+1}, & \hbox{when $-k-\rho<\epsilon(k)\le-k$ with $0<\rho<1$;} \\ \frac{1}{\alpha+2}, & \hbox{when $\epsilon(k)=-k-1+\frac{1}{k}$.} \end{array} \right. \end{align*} So the bigger the parameter function $\epsilon(k_n)$ is, the larger the size of $E_{\epsilon}(\alpha)$ becomes.

Keywords: GCF$_\epsilon$ expansion, Engel series expansion, parameter function, growth rates, Hausdorff dimension

MSC numbers: 11K50, 28A80