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An Inhomogeneous Jarník type theorem for planar curves

Published online by Cambridge University Press:  09 September 2016

DZMITRY BADZIAHIN ,
STEPHEN HARRAP  and
MUMTAZ HUSSAIN
DZMITRY BADZIAHIN
Affiliation:
Durham University, Dept. of Mathematical Sciences, South Rd, Durham, DH1 3LE. e-mail: dzmitry.badziahin@durham.ac.uk, s.g.harrap@durham.ac.uk
STEPHEN HARRAP
Affiliation:
Durham University, Dept. of Mathematical Sciences, South Rd, Durham, DH1 3LE. e-mail: dzmitry.badziahin@durham.ac.uk, s.g.harrap@durham.ac.uk
MUMTAZ HUSSAIN
Affiliation:
The University of Newcastle, School of Mathematical and Physical Sciences, Callaghan, NSW 2308, Australia. e-mail: mumtaz.hussain@newcastle.edu.au
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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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 47 - 70
Copyright
Copyright © Cambridge Philosophical Society 2016 

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