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A Jarník type theorem for planar curves: everything about the parabola

Published online by Cambridge University Press:  01 April 2015

MUMTAZ HUSSAIN
MUMTAZ HUSSAIN*
Affiliation:
School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia. e-mail: mumtaz.hussain@newcastle.edu.au
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Abstract

The well-known theorems of Khintchine and Jarník in metric Diophantine approximation provide a 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.

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Type
Research Article
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Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 1 , July 2015 , pp. 47 - 60
Copyright
Copyright © Cambridge Philosophical Society 2015 

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