HOW THE PARAMETER ε INFLUENCE THE GROWTH RATES OF THE PARTIAL QUOTIENTS IN GCFε EXPANSIONS

For generalized continued fraction (GCF) with parameter ${\epsilon}(k)$, we consider the size of the set whose partial quotients increase rapidly, namely the set $$E_{\epsilon}({\alpha}):=\{x{\in}(0,1]:k_{n+1}(x){\geq}k_n(x)^{\alpha}\;for\;all\;n{\geq}1\}$$, where ${\alpha}$ > 1. We in [6] have obtained the Hausdorff dimension of $E_{\epsilon}({\alpha})$ when ${\epsilon}(k)$ is constant or ${\epsilon}(k){\sim}k^{\beta}$ for any ${\beta}{\geq}1$. As its supplement, now we show that: $$dim_H\;E_{\epsilon}({\alpha})=\{\frac{1}{\alpha},\;when\;-k^{\delta}{\leq}{\epsilon}(k){\leq}k\;with\;0{\leq}{\delta}<1;\\\;\frac{1}{{\alpha}+1},\;when\;-k-{\rho}<{\epsilon}(k){\leq}-k\;with\;0<{\rho}<1;\\\;\frac{1}{{\alpha}+2},\;when\;{\epsilon}(k)=-k-1+\frac{1}{k}$$. So the bigger the parameter function ${\epsilon}(k_n)$ is, the larger the size of $E_{\epsilon}({\alpha})$ becomes.