A note on simultaneous Diophantine approximation on planar curves

Let $\cS_n(\psi_1,...,\psi_n)$ denote the set of simultaneously $(\psi_1,...,\psi_n)$--approximable points in $\R^n$ and $\cSM_n(\psi)$ denote the set of multiplicatively $\psi$--approximable points in $\R^n$. Let $\cM$ be a manifold in $\R^n$. The aim is to develop a metric theory for the sets $\cM \cap \cS_n(\psi_1,...,\psi_n)$ and $\cM \cap \cSM_n(\psi)$ analogous to the classical theory in which $\cM$ is simply $\R^n$. In this note, we mainly restrict our attention to the case that $\cM$ is a planar curve $\cC$. A complete Hausdorff dimension theory is established for the sets $\cC \cap \cS_2(\psi_1,\psi_2)$ and $\cC \cap \cSM_2(\psi)$. A divergent Khintchine type result is obtained for $\cC \cap \cS_2(\psi_1,\psi_2)$; i.e. if a certain sum diverges then the one--dimensional Lebesgue measure on $\cC$ of $\cC \cap \cS_2(\psi_1,\psi_2)$ is full. Furthermore, in the case that $\cC$ is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for $\cC \cap \cS_2(\psi_1,\psi_2)$ naturally generalize the dimension and Lebesgue measure statements of \cite{BDV03}. Within the multiplicative framework, our results for $\cC \cap \cSM_2(\psi)$ constitute the first of their type.