# A note on simultaneous and multiplicative Diophantine approximation on planar curves

BADZIAHIN, D. and LEVESLEY, J. (2007) A note on simultaneous and multiplicative Diophantine approximation on planar curves. Glasgow Mathematical Journal, 49 (2). pp. 367-375. ISSN 1469-509X

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## 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.

Item Type: Article The University of York > Mathematics (York) York RAE Import 12 Jun 2009 10:23 12 Jun 2009 10:23 http://dx.doi.org/10.1017/S0017089507003722 Published Nature Publishing Group 10.1017/S0017089507003722 http://eprints.whiterose.ac.uk/id/eprint/6095