A note on simultaneous Diophantine approximation on planar curves

Let Sn(psi(1),...,.psi(n)) denote the set of simultaneously (psi(1),...,psi(n))approximable points in R-n and S-n*(psi) denote the set of multiplicatively.- approximable points in R-n. Let M be amanifold in R-n. The aim is to develop a metric theory for the sets M boolean AND S-n(psi(1),...,psi(n)) and M boolean AND S-n*(psi) analogous to the classical theory in which M is simply R-n. In this note, we mainly restrict our attention to the case that M is a planar curve C. A complete Hausdorff dimension theory is established for the sets C boolean AND S-2(psi(1),psi(2)) and C n S-2* (psi). A divergent Khintchine type result is obtained for C n S-2(psi(1),psi(2)); i. e. if a certain sum diverges then the one- dimensional Lebesgue measure on C of C boolean AND S-2(psi(1),psi(2)) is full. Furthermore, in the case that C is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for C boolean AND S2(psi(1),psi(2)) naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. ( Mem AMS, 179 ( 846), 1 - 91 ( 2006)). Moreover, within the multiplicative framework, our results for C boolean AND S-2* (psi) constitute the first of their type.