Diophantine approximation in Kleinian groups: singular, extremal, and bad limit points

TL;DR

This paper develops a new 'manifold' framework for metric Diophantine approximation on Kleinian group limit sets, exploring singular, extremal, and badly approximable points, and proposing conjectures related to Khintchine-type results.

Contribution

It introduces a novel 'manifold' theory for Diophantine approximation in Kleinian groups, extending classical concepts to limit sets and proposing new conjectural results.

Findings

Characterization of singular and extremal limit points

Analysis of badly approximable limit points

Discussion of potential Khintchine-type theorems

Abstract

The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural "manifold" strengthening of Sullivan's logarithmic law for geodesics.