A Jarn\'ik type theorem for planar curves: everything about the parabola
Number Theory
2015-05-27 v2
Abstract
The well known theorems of Khintchine and Jarn\'ik in metric Diophantine approximation provide comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develop the theory for planar curves by completing the theory in the case of parabola. This represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.
Keywords
Cite
@article{arxiv.1309.7112,
title = {A Jarn\'ik type theorem for planar curves: everything about the parabola},
author = {Mumtaz Hussain},
journal= {arXiv preprint arXiv:1309.7112},
year = {2015}
}
Comments
13 pages. Revised version