Jarnik-type Inequalities

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.

Comments are welcome! Corrections and modifications in new version

References in corpus (4)

• Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
• Badly approximable vectors on a vertical Cantor set
• Schmidt Games and Conditions on Resonant Sets
• Aperiodic Sequences and Aperiodic Geodesics

Cited by in corpus (4)

• Schmidt Games and Conditions on Resonant Sets
• Diophantine approximations for translation surfaces and planar resonant sets
• Transfer operators and dimension of bad sets for non-uniform Fuchsian lattices
• Dimension Functions on the Spectrum over Bounded Geodesics and Applications to Diophantine Approximation