How the dimension of some GCFϵ sets change with proper choice of the parameter function ϵ(k)

It is interesting that when we choose ϵ(k)=−k, Eϵ(a,b) has the same size with the similar set of regular continued fractions; and when −kρ≤ϵ(k)≤k and ρ<1, Eϵ(a,b) has the same size with the similar set of Engel series.