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Equidistribution of expanding translates of curves and Diophantine approximation on matrices

Abstract

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space \(G/\Gamma \) of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety G / P, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of \(m\times n\) real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet’s theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner’s theorem on measure rigidity for unipotent flows, and linearization technique.

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Fig. 1

Notes

  1. 1.

    The name comes from the notion of stability in geometric invariant theory, and should not be confused with unstable manifolds for a diffeomorphism.

  2. 2.

    It is “minimal” in Kempf’s original definition. Since we are taking limit as t tends to \(\infty \) instead of 0, our numerical function is actually opposite to Kempf’s.

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Acknowledgements

I would like to express my deep gratitude to my advisor Nimish Shah for suggesting this problem, for generously sharing his ideas, and for numerous helpful discussions and constant encouragement. I would like to thank Manfred Einsiedler and Alex Eskin for drawing [15, 16] to my attention. I would like to thank David Anderson for many helpful discussions and his course on equivariant cohomology, where I learned a lot about Schubert varieties. Thanks are due to Menny Aka, Jayadev Athreya, Asaf Katz, Shi Wang, Barak Weiss and Runlin Zhang for helpful discussions and suggestions. Special thanks to Osama Khalil and Dmitry Kleinbock for discussions which helped me to find out a mistake in an earlier version of this paper. Thanks are due to the referee who gave lots of helpful comments. I would like to thank my wife, Yushu Hu, for her unconditional support.

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Yang, P. Equidistribution of expanding translates of curves and Diophantine approximation on matrices. Invent. math. 220, 909–948 (2020). https://doi.org/10.1007/s00222-019-00945-7

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Mathematics Subject Classification

  • 22E40
  • 14L24
  • 11J83