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 is an element of (0,1] : k(n+1)(x) >= k(n)(x)(alpha) for all n >= 1}, where alpha > 1. We in [6] have obtained the Hausdorff dimension of E-epsilon(alpha) when epsilon(k) is constant or epsilon(k) similar to k(beta) for any beta >= 1. As its supplement, now we show that: dim(H) E-epsilon(alpha) = {1/alpha, when -k(delta) <= epsilon(k) <= k with 0 <= delta < 1; 1/alpha+1, when -k - rho < epsilon(k) <= -k with 0 < rho < 1; 1/alpha+2, when epsilon(k) = -k - 1 + 1/k. So the bigger the parameter function epsilon(k(n)) is, the larger the size of E-epsilon(alpha) becomes.