Khintchine's theorem and Diophantine approximation on manifolds

In this paper we initiate a new approach to studying approximations by rational points to smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of $\mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff dimension of $\tau$-well approximable points lying on any nondegenerate submanifold for a range of Diophantine exponents $\tau$ close to $1/n$. Our approach uses geometric and dynamical ideas together with a new technique of `major and minor arcs'. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near `major arcs' and give explicit exponentially small bounds for the measure of `minor arcs'. The latter uses a result of Bernik, Kleinbock and Margulis.