We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of \(\mathbb{R}^{n}\). We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form.

我们证明了 Kleinbock 的猜想，它给出了\(\mathbb{R}^{n}\)的所有极值仿射子空间的清晰分类。我们还给出了所有 Khintchine 型仿射子空间的基本完整的分类，除了两个对数尺度内的一些边界情况。更一般的 Jarník 型定理也得到了证明，有时没有近似函数的单调性。这些结果是我们根据仿射子空间定义矩阵的丢番图性质对接近仿射子空间的有理点数量进行新颖估计的结果。我们的主要工具是多维大筛不等式及其对偶形式。