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A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES

Published online by Cambridge University Press:  09 August 2007

DZMITRY BADZIAHIN  and
JASON LEVESLEY
DZMITRY BADZIAHIN
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom e-mails: db528@york.ac.uk, jl107@york.ac.uk
JASON LEVESLEY
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom e-mails: db528@york.ac.uk, jl107@york.ac.uk
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Abstract

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Let $\mathbb C$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in $\mathbb R^2$ with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

Keywords

Primary 11J83Secondary 11J1311K60

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 2 , May 2007 , pp. 367 - 375
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

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