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Extremal numbers and multi-parametric geometry of numbers

Damien Roy

TL;DR

This work addresses weighted simultaneous rational approximation to points of the form $(1,\xi,\xi^2)$ for extremal $\xi$ within multi-parametric geometry of numbers. The authors develop a combinatorial framework built from Fibonacci words, define a two-parameter, three-coordinate system $\mathbf{P}$, and prove that the trajectory $\mathbf{L}_{\boldsymbol{\xi}}$ is uniformly approximable by a scaled version of $\mathbf{P}$ for an extremal $\boldsymbol{\xi}=(1,\xi,\xi^2)$. Key contributions include a precise partition of the domain, affine behavior of $\mathbf{P}$ on cells, and detailed recurrence relations among approximation points that yield sharp estimates for the exponents of approximation. The results provide explicit asymptotics and Lipschitz-type controls that lead to exact weighted exponents in this extremal setting, illustrating a structured yet intricate interaction between Diophantine approximation, combinatorics of words, and geometry of numbers. The approach yields concrete examples of $\mathbf{L}_{\boldsymbol{\xi}}$-to-$\mathbf{P}$ correspondence with bounded error and supplies a methodology to compute (or bound) exponents of weighted approximation for these points.

Abstract

We study weighted simultaneous rational approximation to points of the form $(1,ξ,ξ^2)$, for a class of extremal real numbers $ξ$, within the framework of multi-parametric geometry of numbers.

Extremal numbers and multi-parametric geometry of numbers

TL;DR

This work addresses weighted simultaneous rational approximation to points of the form for extremal within multi-parametric geometry of numbers. The authors develop a combinatorial framework built from Fibonacci words, define a two-parameter, three-coordinate system , and prove that the trajectory is uniformly approximable by a scaled version of for an extremal . Key contributions include a precise partition of the domain, affine behavior of on cells, and detailed recurrence relations among approximation points that yield sharp estimates for the exponents of approximation. The results provide explicit asymptotics and Lipschitz-type controls that lead to exact weighted exponents in this extremal setting, illustrating a structured yet intricate interaction between Diophantine approximation, combinatorics of words, and geometry of numbers. The approach yields concrete examples of -to- correspondence with bounded error and supplies a methodology to compute (or bound) exponents of weighted approximation for these points.

Abstract

We study weighted simultaneous rational approximation to points of the form , for a class of extremal real numbers , within the framework of multi-parametric geometry of numbers.
Paper Structure (23 sections, 65 theorems, 266 equations, 10 figures)

This paper contains 23 sections, 65 theorems, 266 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Basic tool and preliminary observations
  3. Notation and main results
  4. Monoid of words
  5. Fibonacci sequences
  6. The sets $\mathcal{V}_\ell$
  7. The functions $P_v$
  8. The map ${\mathbf{P}}$
  9. Main result
  10. A family of points
  11. Proof of the main result
  12. The maps $\overline{\iota}$ and $\overline{\alpha}$, and the sets $\overline{\mathcal{V}}_\ell$
  13. The maps $\iota$ and $\alpha$, and the sets $\mathcal{V}_\ell$
  14. A partition of the domain of ${\mathbf{P}}$
  15. Additional properties of the map ${\mathbf{P}}$
  16. ...and 8 more sections

Key Result

Lemma 2.1

For any $\mathbf{q},\mathbf{q}'\in\mathbb{R}^2$, we have $\|\space\mathbf{L}_{\boldsymbol{\xi}}(\mathbf{q})-\mathbf{L}_{\boldsymbol{\xi}}(\mathbf{q}')\space\| \le \|\space\mathbf{q}-\mathbf{q}'\space\|$.

Figures (10)

  • Figure 1: The map $L_\mathbf{x}$ in the generic case.
  • Figure 2: The map $\mathbf{L}_{\boldsymbol{\xi}}$ up to bounded difference outside of ${\mathcal{D}}$, when $\xi_1$ and $\xi_2$ are badly approximable
  • Figure 3: The function $P_v$ attached to a prefix $v$ of $w_\infty$.
  • Figure 4: Partition of ${\mathcal{A}}(\epsilon)$ into admissible polygons for $q_1\le 26$.
  • Figure 5: Partition of $\mathrm{Trap}(u,w)$ when $\alpha(w)<\alpha(u)$
  • ...and 5 more figures

Theorems & Definitions (127)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 117 more