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Counting rational approximations on rank one flag varieties

René Pfitscher

Abstract

On a generalized flag variety of rank one, we count rational approximations to a real point chosen randomly according to the Riemannian volume. In particular, our results apply to Grassmann varieties and quadric hypersurfaces. The proof uses exponential mixing in the space of lattices and tools from geometry of numbers.

Counting rational approximations on rank one flag varieties

Abstract

On a generalized flag variety of rank one, we count rational approximations to a real point chosen randomly according to the Riemannian volume. In particular, our results apply to Grassmann varieties and quadric hypersurfaces. The proof uses exponential mixing in the space of lattices and tools from geometry of numbers.

Paper Structure

This paper contains 14 sections, 12 theorems, 130 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Main results
  4. Method
  5. Notation and conventions
  6. Height and distance
  7. Height on $\mathbf{X}(\mathbb{Q})$
  8. Distance on $X$
  9. Diophantine approximation and counting lattice points
  10. The cone above $X$, coordinates, and measures
  11. Counting lattice points
  12. Diagonal action and well roundedness
  13. From exponential mixing to counting
  14. Counting with respect to the Diophantine exponent

Key Result

Theorem 1.1

Let $X_Q$ be a nonsingular rational quadric hypersurface that is not rationally isomorphic to $X_0$ and contains a rational point. Let $\tau \in (0,1]$. Then there exists a constant $\varkappa > 0$ such that for $\sigma_{Q}$-a.e. $x \in X_Q$,

Figures (2)

  • Figure 1: The set $k_x \mathcal{E}_T$ for the group $G = \operatorname{SL}_2(\mathbb{R})$, $X = \mathbb{P}^1(\mathbb{R})$, $\mathcal{V}_{\chi} = \mathbb{R}^2 \setminus \{0\}$, and $\mathcal{L}_{\chi} = \mathbb{Z}_{\mathrm{pr}}^2$.
  • Figure 2: The action of $a_{y} = y^{-1/2}y^{1/2}$ on $\mathcal{V}_{\chi} = \mathbb{R}^2 \setminus \{0\}$ contracts the line through $\bm{e}_\chi = \bm{e}_1$ and expands the line through $\bm{e}_2$. We show that for a suitable time $y_T$, the set $\mathcal{B}_{T} = a_{y_T} \mathcal{F}_{T}$ is well-rounded and the lattice $a_{y_T} k_x^{-1} \mathbb{Z}^2$ is not too distorted for $\sigma_X$-a.e. $x \in X$. This suggests that $\# \left (a_{y_T} k_x^{-1} \mathcal{L}_{\chi} \cap \mathcal{B}_T \right )$ is (up to scalars) approximately given by the volume of $\mathcal{B}_T$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 3.1
  • proof
  • ...and 16 more