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Extremal Affine Subspaces and Khintchine-Jarník Type Theorems

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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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Notes

  1. They actually only proved the divergence case, but their argument can be used to prove the convergence case as well in a straightforward manner.

  2. An 11th grader at the Davidson Academy when the paper was written.

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Acknowledgements

The author would like to thank Professor John Friedlander for kind help, generous support, and constant encouragement at an early stage of the author’s career, when it simply matters the most! The author is also very grateful to his thesis advisor Professor Robert Vaughan for introducing him to the wonderland of metric diophantine approximation. The author is indebted to the anonymous referee for carefully reading the manuscript and providing helpful comments and suggestions.

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  1. Department of Mathematics and Statistics, University of Nevada, Reno, 1664 N. Virginia St., Reno, NV, 89557, USA

    Jing-Jing Huang

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  1. Jing-Jing Huang
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Correspondence to Jing-Jing Huang.

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Dedicated to Professor John Friedlander

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Research is partially supported by the Simons Foundation Mathematics and Physical Sciences-Collaboration Grants for Mathematicians 855860.

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Huang, JJ. Extremal Affine Subspaces and Khintchine-Jarník Type Theorems. Geom. Funct. Anal. 34, 113–163 (2024). https://doi.org/10.1007/s00039-024-00665-y

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