Convergence results for simultaneous and multiplicative Diophantine approximation on planar curves

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.