Diophantine approximations, large intersections and geodesics in negative curvature

TL;DR

This paper develops new quantitative methods for geodesic approximation in negatively curved spaces, introducing a general Diophantine approximation theorem and exploring measure and dimension phenomena.

Contribution

It presents a novel, flexible framework for geodesic approximation in variable negative curvature, extending classical results and establishing large intersection properties.

Findings

New Diophantine approximation theorem for negatively curved spaces

Extension of logarithm laws and spiraling phenomena analysis

Large intersection property established in this geometric context

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 Jarn\'{i}k-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.