论文标题
极端仿射子空间和khintchine-jarník类型定理
Extremal affine subspaces and Khintchine-Jarník type theorems
论文作者
论文摘要
我们证明了Kleinbock的猜想，该猜想对$ \ Mathbb {r}^n $的所有极端仿射子空间进行了明确的分类。除了两个对数尺度内的某些边界案例外，我们还对所有Khintchine型仿期子空间进行了基本完整的分类。更通用的jarník型定理也被证明，有时没有近似函数的单调性。这些结果是我们新颖的估计值对仿射子空间的合理点数量的后果，其定义矩阵的二聚体特性。我们的主要工具是多维大筛不平等及其双重形式。
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.
