Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves

Details Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves

2006-04-29

arxiv.org/abs/math/0605004v1

Mathematics

Number Theory

13 pages

Scientific paper

Let $\mathcal{C}$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation 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 $\| q x \| < \psi_1(q)$ and $\| qy \| < \psi_2(q)$ infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for the cae of multiplicative approximation $\| qx \| \| q y \| < \psi(q)$, 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.

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