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

In this paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M\subset {\mathbb{R}}^n$ is of dimension strictly greater than $(n+1)\mathrm{/}2$ 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.