An Inhomogeneous Jarník type theorem for planar curves

Badziahin, D.; Harrap, S.; Hussain, M.

Abstract

In metric Diophantine approximation there are classically four main classes of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarník are fundamental to each of them. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximation on manifolds for each of the classes above. In particular, both Khintchine and Jarník-type results have been established for approximation on planar curves except for only one case. In this paper, we prove an inhomogeneous Jarník type theorem for convergence on planar curves in the setting of dual approximation and in so doing complete the metric theory of Diophantine approximation on planar curves.

Citation

Badziahin, D., Harrap, S., & Hussain, M. (2017). An Inhomogeneous Jarník type theorem for planar curves. Mathematical Proceedings of the Cambridge Philosophical Society, 163(1), 47-70. https://doi.org/10.1017/s0305004116000712

• Journal Article Type: Article
• Online Publication Date: Sep 9, 2016
• Publication Date: Jul 1, 2017
• Deposit Date: Dec 30, 2015
• Publicly Available Date: Mar 9, 2017
• Journal: Mathematical Proceedings of the Cambridge Philosophical Society
• Print ISSN: 0305-0041
• Electronic ISSN: 1469-8064
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 163
• Issue: 1
• Pages: 47-70
• DOI: https://doi.org/10.1017/s0305004116000712
• Public URL: https://durham-repository.worktribe.com/output/1416057
• Related Public URLs: http://arxiv.org/abs/1503.04981