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Equidistribution in homogeneous spaces and Diophantine approximation

Yang, Pengyu
http://rave.ohiolink.edu/etdc/view?acc_num=osu1554487183168684

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
In this dissertation, we study the dynamical behavior of translates of submanifolds in homogeneous spaces, and deduce its applications to Diophantine approximation. The study of equidistribution problems in homogeneous dynamics are significant for its usefulness in applications to number theory and geometry. This work is comprised of two main parts. In the first part, we study 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, the Dirichlet's approximation theorem cannot be improved. In the second part, we restrict our discussion to the special linear group, but consider the more general class of differentiable submanifolds. Given a nondegenerate differentiable submanifold of the space of unimodular lattices, we prove that the translates of shrinking balls around a generic point under a diagonal flow get equidistributed with respect to the Haar measure. This implies non-improvability of Dirichlet's approximation theorem for almost every point on a nondegenerate differentiable submanifold of ${\R}^n$, answering a question of Davenport and Schmidt in 1969.
Nimish Shah (Advisor)
104 p.
Mathematics

Recommended Citations

Citations

  • Yang, P. (2019). Equidistribution in homogeneous spaces and Diophantine approximation [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1554487183168684

    APA Style (7th edition)

  • Yang, Pengyu. Equidistribution in homogeneous spaces and Diophantine approximation. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1554487183168684.

    MLA Style (8th edition)

  • Yang, Pengyu. "Equidistribution in homogeneous spaces and Diophantine approximation." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1554487183168684

    Chicago Manual of Style (17th edition)

osu1554487183168684
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This open access ETD is published by The Ohio State University and OhioLINK.