Jarnik-type Inequalities
Dynamical Systems
2017-05-17 v2 Metric Geometry
Number Theory
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.
Cite
@article{arxiv.1306.1314,
title = {Jarnik-type Inequalities},
author = {Steffen Weil},
journal= {arXiv preprint arXiv:1306.1314},
year = {2017}
}
Comments
Comments are welcome! Corrections and modifications in new version