Simultaneous Diophantine Approximation and Asymptotic Formulae on Manifolds Original Research Article

Letψ(r),r=1, 2, … be a positive decreasing sequence such that ∑r=1∞ ψ(r)kdiverges. 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[formula]which holds for almost all points (x1, …, xk) on a smoothm-dimensional submanifoldMof imagek. The manifold satisfies certain curvature conditions which entail restrictions on the codimension. This result extends the known result when the 0?points are not constrained to lie in a submanifold, (i.e., whenM=imagek) to a reasonably general class of manifolds.