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Badly approximable points on manifolds and unipotent orbits in homogeneous spaces

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In this paper, we study the weighted n-dimensional badly approximable points on manifolds. Given a \(C^n\) differentiable non-degenerate submanifold \({\mathcal {U}} \subset {\mathbb {R}}^n\), we will show that any countable intersection of the sets of the weighted badly approximable points on \({\mathcal {U}}\) has full Hausdorff dimension. This strengthens a result of Beresnevich (Invent Math 202(3):1199–1240, 2015) by removing the condition on weights and weakening the smoothness condition on manifolds. Compared to the work of Beresnevich, our approach relies on homogeneous dynamics. It turns out that in order to solve this problem, it is crucial to study the distribution of long pieces of unipotent orbits in homogeneous spaces. The proof relies on the linearization technique and representations of \(\mathrm {SL}(n+1,{\mathbb {R}})\).

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Acknowledgements

The author would like to thank Elon Lindenstrauss and Barak Weiss for sharing many insightful ideas on this problem. He also thanks Shahar Mozes for helpful conversations on this problem. He appreciates their encouragements during the process of this work. He thanks Victor Beresnevich for inspiring discussion on this topic, especially for pointing out that the proof works for \(C^n\) differentiable submanifolds. He also thanks Jinpeng An, Anish Ghosh, Erez Nesharim and Sanju Velani for their interests and helpful comments on an earlier version of this paper. Thanks are due to the anonymous referees for carefully reading the paper and giving many valuable suggestions that led to this revised version.

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  1. College of Mathematics, Sichuan University, Chengdu, 610065, Sichuan, China

    Lei Yang

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The author is supported in part by ISF Grant 2095/15, ERC Grant AdG 267259, NSFC Grant 11743006 and the Fundamental Research Funds for the Central Universities YJ201769.

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Yang, L. Badly approximable points on manifolds and unipotent orbits in homogeneous spaces. Geom. Funct. Anal. 29, 1194–1234 (2019). https://doi.org/10.1007/s00039-019-00508-1

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