Asymptotic statistics for finite continued fractions with restricted digits

TL;DR

This paper provides asymptotic estimates for fractal sets related to bounded partial quotients in continued fractions, offering insights into Zaremba's conjecture and extending to complex cases over quadratic fields.

Contribution

It introduces asymptotic estimates for fractal sets of bounded-type continued fractions and discusses a generalization to complex continued fractions over quadratic fields.

Findings

Asymptotic estimates support an averaging perspective on Zaremba's conjecture.

Results extend to complex continued fractions over imaginary quadratic fields.

Provides a framework for understanding the distribution of bounded partial quotients.

Abstract

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.