Dimension of level sets in GCF expansion with parameters

Let kn(x) be the n-th partial quotient of the generalized continued fraction (GCF) expansion of x. This paper is concerned with the growth rate of kn(x). When the parameter function satisfies −1<ϵ(k)≤1, we obtain the Hausdorff dimension of the setsEϕ={x∈(0,1):limn→∞⁡log⁡kn(x)ϕ(n)=1} for any nondecreasing ϕ with limn→∞⁡(ϕ(n+1)−ϕ(n))=∞ and limn→∞⁡ϕ(n+1)/ϕ(n)=1. Applications are given to several kinds of exceptional sets related to the GCF expansion.