Extremal Affine Subspaces and Khintchine-Jarník Type Theorems

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of . 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...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 34; no. 1; pp. 113 - 163
Main Author Huang, Jing-Jing
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.02.2024
Springer Nature B.V
Subjects
Analysis
Classification
Mathematics
Mathematics and Statistics
Subspaces
Theorems
11J83
11J13
11P21
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Summary:We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of . 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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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-024-00665-y