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Sharpening Vahlen's result in Diophantine approximation

Ayreena Bakhtawar, Cor Kraaikamp

TL;DR

The paper tackles sharpening Vahlen's classical bound for two consecutive approximation coefficients in Diophantine approximation by leveraging Nakada's natural extension of the regular continued fraction. It uses a geometric partition of the natural-extension state space, with bounds that depend on local information from neighboring partial quotients rather than solely on global denominators. In the easy case where $(a_n,a_{n+1})\neq(1,1)$, explicit improved bounds are given in terms of $m=\min\{a_n,a_{n+1}\}$ and $M=\max\{a_n,a_{n+1}\}$; in the difficult case $(1,1)$, the authors show how $a_{n-1}$ and $a_{n+2}$ refine the bounds via intersections of regions in the $\Delta$-plane, yielding concrete inequalities for $\min\{\Theta_{n-1},\Theta_n\}$ and $\max\{\Theta_{n-1},\Theta_n\}$. Overall, the results improve on Vahlen (and extend Borel-type bounds) through a local-quotient, geometry-driven approach, with potential for application to other continued-fraction algorithms such as NICF, OCF, Minkowski, and Rosen fractions.

Abstract

n this paper we refine Vahlen's 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients, especially when at least one of the partial quotients $a_n$ or $a_{n+1}$ of the regular continued fraction expansion $[a_0;a_1,a_2,\dots]$ of $x$ is 1. An improvement of Vahlen's result was already given in papers by Jaroslav Hanucl ([9]), Hanucl and Silvie Bahnerova ([10]), and by Dinesh Sharma Bhattarai ([5]), but the approach of the present paper is very different from Hanucl c.s. We believe that the geometrical methods used in this paper not only offer a significant improvement over Vahlen's result, but also yield new insights that can contribute to improving Borel's classical constant.

Sharpening Vahlen's result in Diophantine approximation

TL;DR

The paper tackles sharpening Vahlen's classical bound for two consecutive approximation coefficients in Diophantine approximation by leveraging Nakada's natural extension of the regular continued fraction. It uses a geometric partition of the natural-extension state space, with bounds that depend on local information from neighboring partial quotients rather than solely on global denominators. In the easy case where , explicit improved bounds are given in terms of and ; in the difficult case , the authors show how and refine the bounds via intersections of regions in the -plane, yielding concrete inequalities for and . Overall, the results improve on Vahlen (and extend Borel-type bounds) through a local-quotient, geometry-driven approach, with potential for application to other continued-fraction algorithms such as NICF, OCF, Minkowski, and Rosen fractions.

Abstract

n this paper we refine Vahlen's 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients, especially when at least one of the partial quotients or of the regular continued fraction expansion of is 1. An improvement of Vahlen's result was already given in papers by Jaroslav Hanucl ([9]), Hanucl and Silvie Bahnerova ([10]), and by Dinesh Sharma Bhattarai ([5]), but the approach of the present paper is very different from Hanucl c.s. We believe that the geometrical methods used in this paper not only offer a significant improvement over Vahlen's result, but also yield new insights that can contribute to improving Borel's classical constant.
Paper Structure (6 sections, 3 theorems, 54 equations, 4 figures)

This paper contains 6 sections, 3 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Using the Ito-Nakada-Tanaka natural extension of the RCF
  3. Nakada's natural extension for the RCF
  4. The "easy case" $(a_n,a_{n+1})\neq (1,1)$
  5. The "difficult case" $(a_n,a_{n+1}) = (1,1)$
  6. Some remarks on sharpening Borel's result

Key Result

Theorem 2.1

Let $x=[0;a_1,a_2,\dots, a_n,a_{n+1},\dots]$, $m=\min \{ a_n,a_{n+1}\}\geq 1$, and $M=\max \{ a_n,a_{n+1}\}\geq 2$. We consider the following cases.

Figures (4)

  • Figure 1: The natural extension $\Omega$ of the RCF (left), and its image $\Delta$ under $\Psi$ (right)
  • Figure 2: Left: $V_M\cap H_m$ in $\Omega$. Right: $I_{M,m}=\Psi(V_M\cap H_m)$ in $\Delta$
  • Figure 3: Intersections $I_{i,j}=\Psi(V_i\cap H_j)$ in $\Delta$, for $i,j=1,2,3$
  • Figure 4: $\Psi (V_{1,a})$ stacked "on top of each other" for $a=1,2,3,4$, and part of $H_{1,1}$

Theorems & Definitions (8)

  • Theorem 2.1
  • Remark 1
  • proof
  • Remark 2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof