In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary non-degenerate 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 non-degenerate 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 ‘generic and special parts’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near the generic part of a non-degenerate manifold. In turn, we give an explicit exponentially small bound for the measure of the special part of the manifold. The latter uses a result of Bernik, Kleinbock and Margulis.

中文翻译：

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流形上的辛钦定理和丢番图近似

在本文中，我们提出了一种新方法来研究 $\mathbb{R}^n$ 光滑子流形上有理点到点的近似。我们的主要结果是 $\mathbb{R}^n$ 的任意非简并子流形的收敛 Khintchine 型定理，它解决了丢番图近似理论中长期存在的问题。此外，我们使用 Hausdorff $s$-measures 细化这个结果，从而获得位于任意非简并子流形上的 $\tau$-well 近似点的 Hausdorff 维数的精确值，对于一系列丢番图指数 $\tau$ close至 $1/n$。我们的方法使用几何和动力学思想以及“通用和特殊零件”新技术。尤其，我们为位于非简并流形的通用部分附近的有界高度的有理点的数量建立了尖锐的上限。反过来，我们给出流形特殊部分的测度的显式指数小界。后者使用了伯尼克、克莱因博克和马古利斯的结果。