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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Bounded and divergent trajectories and expanding curves on homogeneous spaces

Author: Osama Khalil
Journal: Trans. Amer. Math. Soc. 373 (2020), 7473-7525
MSC (2010): Primary 37A17, 22F30, 11J83
Published electronically: August 5, 2020
MathSciNet review: 4155214
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Suppose $g_t$ is a $1$-parameter $\mathrm {Ad}$-diagonalizable subgroup of a Lie group $G$ and $\Gamma < G$ is a lattice. We study the dimension of bounded and divergent orbits of $g_t$ emanating from a class of curves lying on leaves of the unstable foliation of $g_t$ on the homogeneous space $G/\Gamma$. We obtain sharp upper bounds on the Hausdorff dimension of divergent on average orbits and show that the set of bounded orbits is winning in the sense of Schmidt (and, hence, has full dimension). The class of curves we study is roughly characterized by being tangent to copies of $\mathrm {SL}(2,\mathbb {R})$ inside $G$, which are not contained in a proper parabolic subgroup of $G$.

We describe applications of our results to problems in Diophantine approximation by number fields and intrinsic Diophantine approximation on spheres. Our methods also yield the following result for lines in the space of square systems of linear forms: suppose $\varphi (s) = sY + Z$ where $Y\in \mathrm {GL}(n,\mathbb {R})$ and $Z\in M_{n,n}(\mathbb {R})$. Then, the dimension of the set of points $s$ such that $\varphi (s)$ is singular is at most $1/2$ while badly approximable points have Hausdorff dimension equal to $1$.

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Additional Information

Osama Khalil
Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1174
MR Author ID: 1310466
ORCID: 0000-0002-0728-2069

Keywords: Height functions, Hausdorff dimension, divergent trajectories, Schmidt games
Received by editor(s): November 7, 2018
Received by editor(s) in revised form: March 25, 2020
Published electronically: August 5, 2020
Article copyright: © Copyright 2020 American Mathematical Society