Extremal Affine Subspaces and Khintchine-Jarník Type Theorems

IF 2.4 1区数学 Q1 MATHEMATICS

Geometric and Functional Analysis Pub Date : 2024-02-01 DOI:10.1007/s00039-024-00665-y

引用次数: 0

Abstract

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.

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极值仿射子空间和Khintchine-Jarník类型定理

摘要我们证明了克莱因博克的一个猜想，它给出了 \(\mathbb{R}^{n}\) 的所有极值仿射子空间的清晰分类。除了两个对数尺度内的一些边界情况之外，我们还给出了所有欣钦内型仿射子空间的基本完整分类。我们还证明了更一般的雅尼克型定理，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二相性质对接近仿射子空间的有理点数量进行新估计的结果。我们的主要工具是多维大筛不等式及其对偶形式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Geometric and Functional Analysis 数学-数学

CiteScore

3.70

自引率

4.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.