HOW THE PARAMETER ε INFLUENCE THE GROWTH RATES OF THE PARTIAL QUOTIENTS IN GCFε EXPANSIONS
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For generalized continued fraction (GCF) with parameter ε(k), we consider the size of the set whose partial quotients increase rapidly, namely the set Eε(α) := {x ∈ 2 (0, 1] : kn+1(x) ≥ kn(x)α for all n ≥ 1}, where α > 1. We in [6] have obtained the Hausdorff dimension of Eε(α) when ε(k) is constant or ε(k) ~ kβ for any β ≥ 1. As its supplement, now we show that: dimH Eε(α) = {[수식] So the bigger the parameter function ε(kn) is, the larger the size of Eε(α) becomes.
Abstract
For generalized continued fraction (GCF) with parameter ε(k), we consider the size of the set whose partial quotients increase rapidly, namely the set Eε(α) := {x ∈ 2 (0, 1] : kn+1(x) ≥ kn(x)α for all n ≥ 1}, where α > 1. We in [6] have obtained the Hausdorff dimension of Eε(α) when ε(k) is constant or ε(k) ~ kβ for any β ≥ 1. As its supplement, now we show that: dimH Eε(α) = {[수식] So the bigger the parameter function ε(kn) is, the larger the size of Eε(α) becomes.
참고문헌 (6)
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- T. Zhong, "The growth rate of the partial quotients in a class of continued fractions with parameters", J. Number Theory, 145, pp.388~401 (2014)