Asymptotic statistics for finite continued fractions with restricted digits

Zaremba’s conjecture concerns a formation of continued fraction expansions for rational numbers with partial quotient bounded by an absolute constant. We present asymptotic estimates for the size of $\epsilon$-thickening of certain fractal sets of bounded-type, which in turn provide a remark on Zaremba’s conjecture in an averaging sense. We also discuss a generalisation for complex continued fractions over imaginary quadratic fields.