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

Title & Authors
HOW THE PARAMETER ε INFLUENCE THE GROWTH RATES OF THE PARTIAL QUOTIENTS IN GCFε EXPANSIONS
Zhong, Ting; Shen, Luming;

Abstract
For generalized continued fraction (GCF) with parameter $\small{{\epsilon}(k)}$, we consider the size of the set whose partial quotients increase rapidly, namely the set $\small{E_{\epsilon}({\alpha}):=\{x{\in}(0,1]:k_{n+1}(x){\geq}k_n(x)^{\alpha}\;for\;all\;n{\geq}1\}}$, where $\small{{\alpha}}$ > 1. We in [6] have obtained the Hausdorff dimension of $\small{E_{\epsilon}({\alpha})}$ when $\small{{\epsilon}(k)}$ is constant or $\small{{\epsilon}(k){\sim}k^{\beta}}$ for any $\small{{\beta}{\geq}1}$. As its supplement, now we show that: $\small{dim_H\;E_{\epsilon}({\alpha})=\{\frac{1}{\alpha},\;when\;-k^{\delta}{\leq}{\epsilon}(k){\leq}k\;with\;0{\leq}{\delta}}$$\small{&}$$\small{lt;1;\\\;\frac{1}{{\alpha}+1},\;when\;-k-{\rho}}$$\small{&}$$\small{lt;{\epsilon}(k){\leq}-k\;with\;0}$$\small{&}$$\small{lt;{\rho}}$$\small{&}$$\small{lt;1;\\\;\frac{1}{{\alpha}+2},\;when\;{\epsilon}(k)=-k-1+\frac{1}{k}}$. So the bigger the parameter function $\small{{\epsilon}(k_n)}$ is, the larger the size of $\small{E_{\epsilon}({\alpha})}$ becomes.
Keywords
$\small{GCF_{\epsilon}}$ expansion;Engel series expansion;parameter function;growth rates;Hausdorff dimension;
Language
English
Cited by
1.
How the dimension of some GCF ϵ sets change with proper choice of the parameter function ϵ ( k ), Journal of Number Theory, 2017, 174, 1
2.
Some dimension relations of the Hirst sets in regular and generalized continued fractions, Journal of Number Theory, 2016, 167, 128
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