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Diophantine approximations, large intersections and geodesics in negative curvature

Anish Ghosh, Debanjan Nandi

Abstract

In this paper we prove quantitative results about geodesic approximations to submanifolds in negatively curved spaces. Among the main tools is a new and general Jarník-Besicovitch type theorem in Diophantine approximation. The framework we develop is flexible enough to treat manifolds of variable negative curvature, a variety of geometric targets, and logarithm laws as well as spiraling phenomena in both measure and dimension aspect. Several of the results are new also for manifolds of constant negative sectional curvature. We further establish a large intersection property of Falconer in this context.

Diophantine approximations, large intersections and geodesics in negative curvature

Abstract

In this paper we prove quantitative results about geodesic approximations to submanifolds in negatively curved spaces. Among the main tools is a new and general Jarník-Besicovitch type theorem in Diophantine approximation. The framework we develop is flexible enough to treat manifolds of variable negative curvature, a variety of geometric targets, and logarithm laws as well as spiraling phenomena in both measure and dimension aspect. Several of the results are new also for manifolds of constant negative sectional curvature. We further establish a large intersection property of Falconer in this context.

Paper Structure

This paper contains 21 sections, 28 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. A correspondence between dynamics of the geodesic flow and Diophantine approximation
  3. Large Intersections
  4. Preliminaries
  5. The visual metric
  6. The Patterson-Sullivan (quasiconformal) measure and Ahlfors regularity
  7. Dyadic and Whitney decompositions
  8. Hausdorff content and dimension
  9. CAT($-1$) spaces
  10. Dynamics of the $\Gamma$-action on the boundary
  11. Diophantine Approximation
  12. Generalized Jarnik-Besicovitch
  13. Large intersections
  14. A Borel-Cantelli lemma
  15. Summary
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $M$ be a closed manifold of dimension $n$ with constant negative sectional curvature, $k=-1$. Let $N$ be a compact, totally geodesic submanifold of $M$ of dimension $0\leq s\leq n-1$. Let $\tau\geq 0$ be fixed. Then given $x_0\in M$, we have that the set has Hausdorff dimension where $\gamma_v$ is the geodesic at $x_0$ at time zero with direction $v$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Shadow
  • Lemma 2.2
  • proof
  • Lemma 2.3: Sullivan's Shadow Lemma
  • Lemma 2.4
  • proof
  • ...and 54 more