Simultaneous diophantine approximation and asymptotic formulae on manifolds

Let ?(r), r =1, 2, ... be a positive decreasing sequence such that ?⁸_r=1 ?(r)^k diverges. Using a powerful variance argument due to Schmidt, an asymptotic formula is obtained for the number of integer solutions q of the system of Diophantine inequalities max{ ||qx_i||: 1 = i = k} < ?(q) which holds for almost all points (x₁, ..., x_k) on a smooth m-dimensional submanifold M of R^k. The manifold satisfies certain curvature conditions which entail restrictions on the codimension. This result extends the known result when the points are not constrained to lie in a submanifold, (i.e., when M = R^k) to a reasonably general class of manifolds. © 1996 Academic Press, Inc.