Asymptotic statistics for finite continued fractions with restricted digits

2025/12/12 by Lee, Jungwon
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #semigroups and automata theory #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2512.11357

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 ε-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.

Citations

• Euclidean algorithms are Gaussian over imaginary quadratic fields
• Complex numbers with a prescribed order of approximation and Zaremba's conjecture
• Congruence counting in Schottky and continued fractions semigroups of\n \SO(n, 1)
• Popular Products and Continued Fractions
• Euclidean algorithms are Gaussian

Related

• On Zaremba's Conjecture
• Fractal geometry of continued fractions with large coefficients and dimension drop problems
• A strengthening of a theorem of Bourgain-Kontorovich-V
• Continued Fractions of Arithmetic Sequences of Quadratics
• On certain statistical properties of continued fractions with even and\n with odd partial quotients