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Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of PSL(2,R)

Zheng, Cheng
http://rave.ohiolink.edu/etdc/view?acc_num=osu1467320625

Abstract Details

2016, Doctor of Philosophy, Ohio State University, Mathematics.
We consider the orbits {pu(n^{1+γ})|n ∈ N} in Γ\PSL(2,R), where Γ is a nonuniform lattice in PSL(2,R) and {u(t)} is the standard unipotent one-parameter subgroup in PSL(2,R). Under a Diophantine condition on the intial point p, we can prove that the trajectory {pu(n^{1+γ})|n ∈ N} is equidistributed in Γ\PSL(2,R) for small γ > 0, which generalizes a result of Venkatesh [V10]. In Chapter 2, we will compute Hausdorff dimensions of subsets of non-Diophantine points in Γ\PSL(2,R), using results of lattice counting problem. In Chapter 3 we will use the exponential mixing property of a semisimple flow to prove the effective equidistribution of horospherical orbits. In Chapter 4, we will give a definition of Diophantine points of type γ for γ ≥ 0 in a homogeneous space Γ\G and compute the Hausdorff dimension of the subset of points which are not Diophantine of type γ when G is a semisimple Lie group of real rank one. As an application, we will deduce a Jarnik-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.
Nimish Shah (Advisor)
Vitaly Bergelson (Committee Member)
James Cogdell (Committee Member)
115 p.
Mathematics

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Citations

  • Zheng, C. (2016). Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of PSL(2,R) [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1467320625

    APA Style (7th edition)

  • Zheng, Cheng. Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of PSL(2,R). 2016. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1467320625.

    MLA Style (8th edition)

  • Zheng, Cheng. "Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of PSL(2,R)." Doctoral dissertation, Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1467320625

    Chicago Manual of Style (17th edition)

osu1467320625
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This open access ETD is published by The Ohio State University and OhioLINK.