Diophantine approximation on planar curves and the distribution of rational points

Let $\mathcal{C}$ be a non–degenerate planar curve and for a real, positive decreasing function $\psi$ let $\mathcal{C}(\mathcal{\psi })$ denote the set of simultaneously $\psi$–approximable points lying on $\mathcal{C}$. We show that $\mathcal{C}$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\mathcal{C}$ of $\mathcal{C}(\mathcal{\psi })$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that $\mathcal{C}$ is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions $\psi$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\mathcal{C}(\mathcal{\psi })$. These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.