Glasgow Mathematical Journal

Research Article

A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES

DZMITRY BADZIAHINa1 and JASON LEVESLEYa1

a1 Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom e-mails: db528@york.ac.uk, jl107@york.ac.uk

Abstract

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.

(Received October 31 2006)

(Revised February 01 2007)

(Accepted March 04 2007)

Key Words:

  • Primary 11J83;
  • Secondary 11J13;
  • 11K60