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 in:Geometric and functional analysis 2024-02, Vol.34 (1), p.113-163
Main Author: Huang, Jing-Jing
Format: Article
Language:English
Subjects:
Analysis
Classification
Mathematics
Mathematics and Statistics
Subspaces
Theorems
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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.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-024-00665-y