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In this paper we initiate a new approach to studying approximations byrational points to points on smooth submanifolds of $\mathbb{R}^n$. Our mainresult is a convergence Khintchine type theorem for arbitrary nondegeneratesubmanifolds of $\mathbb{R}^n$, which resolves a longstanding problem in thetheory of Diophantine approximation. Furthermore, we refine this result usingHausdorff $s$-measures and consequently obtain the exact value of the Hausdorffdimension of $\tau$-well approximable points lying on any nondegeneratesubmanifold for a range of Diophantine exponents $\tau$ close to $1/n$. Ourapproach uses geometric and dynamical ideas together with a new technique of`generic and special parts'. In particular, we establish sharp upper bounds forthe number of rational points of bounded height lying near the generic part ofa non-degenerate manifold. In turn, we give an explicit exponentially smallbound for the measure of the special part of the manifold. The latter uses aresult of Bernik, Kleinbock and Margulis.

Khintchine's theorem and Diophantine approximation on manifolds