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On geometry of simultaneous approximation to three real numbers

Antoine Marnat, Nikolay Moshchevitin

Abstract

Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation, optimal in terms of this geometry.

On geometry of simultaneous approximation to three real numbers

Abstract

Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation, optimal in terms of this geometry.
Paper Structure (14 sections, 10 theorems, 195 equations, 2 figures)

This paper contains 14 sections, 10 theorems, 195 equations, 2 figures.

Table of Contents

  1. Simultaneous approximation and Diophantine exponents
  2. Patterns of best approximation vectors
  3. Parameters
  4. Results
  5. Proof of Theorem 1
  6. Inductive construction for Theorem 2: $k=1$
  7. Base of induction
  8. Primitive vectors
  9. The initial triple $\pmb{z}_1,\pmb{z}_2,\pmb{z}_3$
  10. Two- and three-dimensional lattices
  11. Step of induction
  12. Stage 1
  13. Stage 2
  14. Comments on the case $k>1$

Key Result

Lemma 1

For any $k\geqslant 1$ and for any $\lambda \in \left(\frac{1}{3}, 1\right)$ the equation in the interval $\left( 1,\frac{2}{\theta}\right)$ has a unique root $\frak{g}_k = \frak{g}_k (\lambda)$. This root has lower bound Moreover, for a fixed $\lambda$ one has

Figures (2)

  • Figure 1: Angular domain $\mathcal{U}_j$ and the neighborhood $\mathcal{W}_j$ inside, section by subspace $S$; inequality for angles $\frac{\pi}{2}-o(1) =\angle \pmb{Z}_{j-1}\pmb{X}"\pmb{Z}_j < \angle \pmb{Z}_{j-1}\pmb{X}_j\pmb{Z}_j < \angle \pmb{Z}_{j-1}\pmb{X}'\pmb{Z}_j < \angle \pmb{Z}_{j-1}\pmb{Z}_j\pmb{Z}' = \frac{3\pi}{4}.$
  • Figure 2: To Lemma 4: small ellipse at the left side close to the point $\overline{\pmb{Z}}_{2\nu+3}$ is the neighborhood $\overline{\mathcal{W}}_{2\nu+3}\subset \mathbb{R}_1^3$ with center $\overline{\pmb{X}}_{2\nu+3}$ and the large ellipse at the right side is the image $\frak{P}_1(\overline{\mathcal{W}}_{2\nu+3}) \subset L_1$.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2
  • proof
  • ...and 14 more