Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the spectrum of Diophantine exponents of lattices

Oleg N. German

Abstract

In this paper we describe the spectrum of values of weak uniform Diophantine exponents of lattices in arbitrary dimension.

On the spectrum of Diophantine exponents of lattices

Abstract

In this paper we describe the spectrum of values of weak uniform Diophantine exponents of lattices in arbitrary dimension.
Paper Structure (15 sections, 8 theorems, 128 equations, 1 figure)

This paper contains 15 sections, 8 theorems, 128 equations, 1 figure.

Table of Contents

  1. Diophantine exponents of lattices
  2. Hyperbolic minima of lattices
  3. Two-dimensional case and additional linear forms
  4. Constructing a two-dimensional subspace and a rank $2$ lattice in it
  5. Complementing $\Gamma_\mathcal{L}$ to a lattice of rank $d$
  6. Main metric lemma
  7. Reduction to Borel--Cantelli lemma.
  8. Sets $\mathcal{F}^i_\nu$ and $\mathcal{M}^i_\nu$.
  9. Covering $\mathcal{F}^1_\nu$ by parallelepipeds.
  10. Analysis of $\mathcal{N}(\mathbf{z},\pmb\tau)$
  11. Estimating the number of nonempty $\mathcal{N}^j(\mathbf{z},\pmb\tau)$ for a fixed $\mathbf{z}_E$.
  12. Estimating the measure of $\mathcal{M}_\nu$.
  13. Proof of Theorems \ref{['t:spectrum_of_the_regular_arbitrary_d']} and \ref{['t:spectrum_of_the_weak_arbitrary_d']}
  14. Choosing $\theta$ and $\eta$ to prove Theorem \ref{['t:spectrum_of_the_regular_arbitrary_d']}.
  15. Choosing $\theta$ and $\eta$ to prove Theorem \ref{['t:spectrum_of_the_weak_arbitrary_d']}.

Key Result

Theorem 1

For every $d\geqslant2$, we have $\pmb\Omega_d=[0,+\infty]$.

Figures (1)

  • Figure 1: Successive hyperbolic minima

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark
  • ...and 8 more