  This work is motivated by problems on simultaneous Diophantine approximation
on manifolds, namely, establishing Khintchine and Jarnik type theorems for
submanifolds of R^n. These problems have attracted a lot of interest since
Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G.
Sprindzuk. They have been settled for planar curves but remain open in higher
dimensions. In this paper, Khintchine and Jarnik type divergence theorems are
established for arbitrary analytic non-degenerate manifolds regardless of their
dimension. The key to establishing these results is the study of the
distribution of rational points near manifolds -- a very attractive topic in
its own right. Here, for the first time, we obtain sharp lower bounds for the
number of rational points near non-degenerate manifolds in dimensions n>2 and
show that they are ubiquitous (that is uniformly distributed).
