Skip to Main Content
 

Global Search Box

 
 
 
 

Files

File List


Khalil_Thesis.pdf (831.57 KB)

ETD Abstract Container

Abstract Header

On the Dimension of Certain Divergent Trajectories on Homogeneous Spaces and Diophantine Approximation

Khalil, Osama
http://rave.ohiolink.edu/etdc/view?acc_num=osu156345228437039

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
Motivated by problems in Diophantine approximation, we study the Hausdorff dimension of certain divergent orbits of diagonlizable flows on homogeneous spaces. We develop an approach to these problems relying on Margulis functions and systems of integral inequalities. In Chapter 2, we use our method to provide an alternative proof of the Baker- Sprindzuk conjecture, proved originally by Kleinbock and Margulis. In particular, we show that very well approximable matrices have measure 0 with respect to a parameter measure on a non-degenerate curve. One advantage of this approach is that it bypasses the rather technical general theory of (C, alpha)-good functions. Instead, we use the much simpler (C, alpha)-good property of polynomials. Moreover, we prove a stronger dynamical statement concerning recurrence of translates of shrinking segments of curves, Theorem 1.2.6. In Chapter 3, we provide an upper bound on the dimension of divergent orbits of certain diagonal flows emanating from curves on a class of homogeneous spaces. This class includes quotients of products of R-rank one Lie groups. Our upper bound formula is in terms of quantities measuring the non-degeneracy of the curve. We also demonstrate that our upper bound is optimal in various senses. Moreover, we show that the orbits which remain bounded form a winning set for a game introduced by Wolfgang Schmidt. This, in particular, implies that these bounded orbits have full Hausdorff dimension. These results are contained in Theorems 1.3.1, 1.3.5, and 1.3.6. Our proof involves developing abstract criteria implying our dimension estimates for general actions of Lie groups on topological spaces. We also introduce a general class of curves on homogeneous spaces which satisfy our criteria. We call this general class of curves: deformations of maximal representations. In Chapter 4, we study divergent orbits of a diagonal flow emanating from self-similar fractals on the space of unimodular lattices. Our main result, Theorem 1.4.2, provides a sharp upper bound on the Hausdorff dimension of these orbits in terms of quantities which are closely tied to the Frostman exponents of projections of a canonical Hausdorff measure supported on the fractal. Via Dani's correspondence, this result implies an upper bound on the dimension of singular vectors on fractals. As a corollary, we show that the dimension of singular vectors on the product of 2 copies of Cantor's middle thirds set is at most log 16/log 27, thus addressing the upper bound part of a question raised by Bugeaud, Cheung, and Chevallier. It is worth noting that our result recovers the exact dimension of singular vectors in R^d, obtained by Cheung and Chevallier. Our proof develops an abstract approach to the study of these questions. As a result, we are able to study divergent orbits of the Teichmuller geodesic flow, yielding a sharp upper bound on the dimension of non-uniquely ergodic directions for translation flows on flat surfaces belonging to a fractal.
Nimish Shah (Advisor)
Vitaly Bergelson (Committee Member)
Andrey Gogolyev (Committee Member)
Jean-Francois Lafont (Committee Member)
Mathematics

Recommended Citations

Citations

  • Khalil, O. (2019). On the Dimension of Certain Divergent Trajectories on Homogeneous Spaces and Diophantine Approximation [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu156345228437039

    APA Style (7th edition)

  • Khalil, Osama. On the Dimension of Certain Divergent Trajectories on 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=osu156345228437039.

    MLA Style (8th edition)

  • Khalil, Osama. "On the Dimension of Certain Divergent Trajectories on Homogeneous Spaces and Diophantine Approximation." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu156345228437039

    Chicago Manual of Style (17th edition)

osu156345228437039
408
© 2019, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.