Dimension of level sets in GCF expansion with parameters.

Let k n ( 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 k n ( x ) . When the parameter function satisfies − 1 < ϵ ( k ) ≤ 1 , we obtain the Hausdorff dimension of the sets E ϕ = { x ∈ ( 0 , 1 ) : lim n → ∞ ⁡ log ⁡ k n ( x ) ϕ ( n ) = 1 } for any nondecreasing ϕ with lim n → ∞ ⁡ ( ϕ ( n + 1 ) − ϕ ( n ) ) = ∞ and lim n → ∞ ⁡ ϕ ( n + 1 ) / ϕ ( n ) = 1 . Applications are given to several kinds of exceptional sets related to the GCF expansion. [ABSTRACT FROM AUTHOR]