The growth rate of the partial quotients in a class of continued fractions with parameters

Let ϵ:N→Rϵ:N→R be a parameter function satisfying the condition ϵ(k)+k+1>0ϵ(k)+k+1>0 and let Tϵ:(0,1]→(0,1]Tϵ:(0,1]→(0,1] be a transformation defined byTϵ(x)=−1+(k+1)x1+ϵ(k)−kϵ(k)xfor x∈(1/(k+1),1/k]. Under the algorithm TϵTϵ, every x∈(0,1]x∈(0,1] is attached an expansion, called generalized continued fraction expansion with parameters by F. Schweiger [3]. Define the sequence {kn(x)}n≥1{kn(x)}n≥1 of the partial quotients of x   by k1(x)=⌊1/x⌋k1(x)=⌊1/x⌋ and kn(x)=k1(Tϵn−1(x)) for every n≥2n≥2. It is clear that under the condition satisfied by the parameter function ϵ  , kn+1(x)≥kn(x)kn+1(x)≥kn(x) for all n≥1n≥1. In this paper, we consider the size of the set given byEϵ(α):={x∈(0,1]:kn+1(x)≥kn(x)α for all n≥1}Eϵ(α):={x∈(0,1]:kn+1(x)≥kn(x)α for all n≥1} for any α≥1α≥1. We show thatdimH⁡Eϵ(α)={1α,when ϵ(k)≡ϵ0 (constant);1α−β+1,when ϵ(k)∼kβ and α≥β≥1;1,when ϵ(k)∼kβ and α<β. where dimHdimH denotes the Hausdorff dimension. The first result generalizes a result of J. Wu [5] who considered the case when ϵ≡0ϵ≡0 (i.e., Engel expansion).