Diophantine approximation on manifolds and the distribution of rational points: Contributions to the convergence theory

Abstract

In this article, we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold M ⊂ ℝn is of dimension strictly greater than (n+1)/2 and satisfies a natural non-degeneracy condition, then M is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.