Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A unifying theory for metrical results on regular continued fraction convergents and mediants

Karma Dajani, Cor Kraaikamp, Slade Sanderson

TL;DR

This work unifies the metric theory of regular continued fraction convergents and mediants by reexamining Ito's natural extension of the Farey tent map and employing induced dynamics on carefully chosen subregions. Through relative equidistribution and explicit geometric encodings of approximation coefficients, it derives limiting distributions for subsequences, recovers classical Legendre–type refinements and Fatou–Grace–Koksma insights, and generalizes Lévy-type theorems to Farey-convergent subsequences. The framework yields both streamlined proofs of known results and new Lévy-type statements, Doeblin–Lenstra-type distributions, and detailed behavior of consecutive convergents and mediants, all within a coherent dynamical-systems picture. The approach relies on induced maps and two-dimensional representations to connect forward orbits with the full Farey/convergent structure, providing clear geometric interpretations and broad applicability to metric Diophantine approximation in continued fractions.

Abstract

We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results -- including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of Lévy's Theorem to subsequences of convergents and mediants -- are presented as corollaries within this unifying theory.

A unifying theory for metrical results on regular continued fraction convergents and mediants

TL;DR

This work unifies the metric theory of regular continued fraction convergents and mediants by reexamining Ito's natural extension of the Farey tent map and employing induced dynamics on carefully chosen subregions. Through relative equidistribution and explicit geometric encodings of approximation coefficients, it derives limiting distributions for subsequences, recovers classical Legendre–type refinements and Fatou–Grace–Koksma insights, and generalizes Lévy-type theorems to Farey-convergent subsequences. The framework yields both streamlined proofs of known results and new Lévy-type statements, Doeblin–Lenstra-type distributions, and detailed behavior of consecutive convergents and mediants, all within a coherent dynamical-systems picture. The approach relies on induced maps and two-dimensional representations to connect forward orbits with the full Farey/convergent structure, providing clear geometric interpretations and broad applicability to metric Diophantine approximation in continued fractions.

Abstract

We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results -- including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of Lévy's Theorem to subsequences of convergents and mediants -- are presented as corollaries within this unifying theory.
Paper Structure (21 sections, 34 theorems, 224 equations, 8 figures)

This paper contains 21 sections, 34 theorems, 224 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background, definitions and notation
  3. (Semi-)regular continued fractions
  4. The Gauss map
  5. The Farey tent map
  6. Lehner expansions
  7. Farey expansions and convergents
  8. Farey expansions
  9. Farey convergents
  10. Ito's natural extension of the Farey tent map
  11. Inducing Ito's natural extension
  12. The (relative) equidistribution of ($\mathcal{F}$-) $\mathcal{F} _R$-orbits
  13. Metrical results
  14. Approximation coefficients and their limiting distributions
  15. On the theorems of Legendre, Fatou--Grace and Koksma
  16. ...and 6 more sections

Key Result

Proposition 3.1

For each $n\ge 0$, where $P_n/Q_n=[b_0-1;e_1/b_1,\dots,e_n/b_n]$ is the $n^{\text{th}}$ Farey convergent of $x$.

Figures (8)

  • Figure 1: The Gauss map $G$ (black) and the Farey tent map $F$ (blue). Both maps coincide on the domain $[1/2,1]$.
  • Figure 2: From left to right: The sets $V_3\cap H_1,\ \mathcal{F} (V_3\cap H_1),\ \mathcal{F} ^2(V_3\cap H_1)$ and $\mathcal{F} ^3(V_3\cap H_1)$, respectively.
  • Figure 3: The curves $h(x,y)=z$ for $z\in\{0,1,2,3,4\}$. The region $S_1$ is the shaded region above the curve $h(x,y)=1$.
  • Figure 4: The regions $R$ considered in Corollaries \ref{['cor.i']}--\ref{['cor.vi']} and \ref{['cor.vii']}.
  • Figure 5: Red and blue regions correspond to negative and positive signatures $\delta(x,u_n/s_n)=-1,1$, respectively. The curves $h(x,y)=z$ are shown in yellow for $z\in\{1/2,2/3,1,2\}$.
  • ...and 3 more figures

Theorems & Definitions (62)

  • Proposition 3.1
  • proof
  • Theorem 4.1: Theorem 1 of BY1996
  • Definition 4.2
  • Remark 4.3
  • Example 4.4
  • Example 4.5
  • Theorem 4.6
  • proof
  • Theorem 4.7: Theorem 2 of BY1996
  • ...and 52 more