In this paper, we complete the long-standing challenge to establish a Khintchine-type theorem for arbitrary nondegenerate manifolds in Rⁿ. In particular, our main result finally removes the analyticity assumption from the Khintchine type theorem proved in Ann. of Math. 175 (2012), 187-235. Furthermore, we obtain a Jarn\'ik-type refinement of our main result, which uses Hausdorff measures. The results are also obtained in the inhomogeneous setting. The proofs are underpinned by a sharper version of a `quantitative nondivergence' result of Bernik--Kleinbock--Margulis and a duality argument which we use to study rational points near manifolds.
Beresnevich et al. (Fri,) studied this question.