Jarnik-type Inequalities

TL;DR

This paper extends Jarnik's inequalities to new mathematical contexts, providing conditions under which Hausdorff-dimension bounds apply to sets of badly approximable points and related geometric objects.

Contribution

It introduces a framework with simple conditions that generalize Jarnik's inequalities to various settings, including weighted badly approximable vectors and hyperbolic geodesics.

Findings

Established conditions for extending Jarnik's inequalities

Applied framework to weighted badly approximable vectors

Analyzed geodesics avoiding convex sets in hyperbolic space

Abstract

It is well known due to Jarnik that the set Bad of badly approximable numbers is of Hausdorff-dimension one. If Bad(c) denotes the subset of x in Bad for which the approximation constant c > c(x), then Jarnik was in fact more precise and gave nontrivial lower and upper bounds of the Hausdorff-dimension of Bad(c) in terms of the parameter c > 0. Our aim is to determine simple conditions on a framework which allow to extend 'Jarnik's inequality' to further examples; among the applications, we discuss the set of badly approximable vectors in with weights and the set of geodesics in the hyperbolic space which avoid a suitable collection of convex sets.