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Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation on square matrices

Author: Lei Yang
Journal: Proc. Amer. Math. Soc. 144 (2016), 5291-5308
MSC (2010): Primary 37A17; Secondary 22F30, 11J13
Published electronically: July 21, 2016
MathSciNet review: 3556272
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Abstract: In this paper, we study an analytic curve $ \varphi : I = [a, b] \rightarrow $
$ \mathrm {M}(n\times n, \mathbb{R})$ in the space of $ n$ by $ n$ real matrices. There is a natural map
$ u : \mathrm {M}(n\times n, \mathbb{R}) \rightarrow H = \mathrm {SL}(2n,\mathbb{R})$. Let $ G$ be a Lie group containing $ H$ and $ \Gamma < G$ be a lattice of $ G$. Let $ X = G/\Gamma $. Then given a dense $ H$-orbit in $ X$, one could embed $ u(\varphi (I))$ into $ X$. We consider the expanding translates of the curve by some diagonal subgroup $ A = \{a(t ) : t \in \mathbb{R}\} \subset H$. We will prove that if $ \varphi $ satisfies certain geometric conditions, then the expanding translates will tend to be equidistributed in $ G/\Gamma $, as $ t \rightarrow +\infty $. As an application, we show that for almost every point on $ \varphi (I)$, the Diophantine approximation given by Dirichlet's Theorem is not improvable.

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Additional Information

Lei Yang
Affiliation: Mathematical Sciences Research Institute, Berkeley, California 94720
Address at time of publication: Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 9190401, Israel

Keywords: Equidistribution, homogeneous spaces, Ratner's Theorem, Diophantine approximation, Dirichlet's Theorem
Received by editor(s): August 11, 2015
Received by editor(s) in revised form: February 19, 2016, February 23, 2016, and February 24, 2016
Published electronically: July 21, 2016
Additional Notes: The author was supported in part by a Postdoctoral Fellowship at MSRI
Communicated by: Nimish Shah
Article copyright: © Copyright 2016 American Mathematical Society