Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

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September 2020, 40(9): 5247-5287. doi: 10.3934/dcds.2020227

Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices

Nimish Shah ^1, and Lei Yang ^2,,

1.	Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
2.	College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

^* Corresponding author: Lei Yang

Received September 2019 Revised February 2020 Published June 2020

Fund Project: This material is based upon work supported by the National Science Foundation under Grant Nos. 1301715 and 1700394.
The second author is supported in part by ISF grant 2095/15, ERC grant AdG 267259, NSFC grant 11743006, 11801384 and the Fundamental Research Funds for the Central Universities YJ201769

In this paper, we study an analytic curve $ \varphi: I = [a, b]\rightarrow \mathrm{M}(m\times n, \mathbb{R}) $ in the space of $ m $ by $ n $ real matrices, and show that if $ \varphi $ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into a homogeneous space $ G/\Gamma $, and prove that under the action of some expanding diagonal subgroup $ A = \{a(t): t \in \mathbb{R}\} $, the translates of the curve tend to be equidistributed in $ G/\Gamma $, as $ t \rightarrow +\infty $. The proof relies on the linearization technique and representation theory.

Keywords: Dirichlet's theorem, Diophantine approximation, homogeneous spaces, equidistribution, Ratner's theorem.

Mathematics Subject Classification: Primary: 11J13, 11J83; Secondary: 22E40, 37A17.

Citation: Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227

References:

[1]	M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, On metric diophantine approximation in matrices and lie groups, C. R. Math. Acad. Sci. Paris, 353 (2015), 185-189. doi: 10.1016/j.crma.2014.12.007.
[2]	M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, Diophantine approximation on matrices and lie groups, Geom. Funct. Anal., 28 (2018), 1-57. doi: 10.1007/s00039-018-0436-0.
[3]	R. C. Baker, Dirichlet's theorem on Diophantine approximation, Math. Proc. Cambridge Philos. Soc., 83 (1978), 37-59. doi: 10.1017/S030500410005427X.
[4]	Y. Bugeaud, Approximation by algebraic integers and hausdorff dimension, Journal of the London Mathematical Society (2), 65 (2002), 547-559. doi: 10.1112/S0024610702003137.
[5]	S. G. Dani, On orbits of unipotent flows on homogeneous spaces, Ergodic Theory and Dynamical Systems, 4 (1984), 25-34. doi: 10.1017/S0143385700002248.
[6]	S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17. doi: 10.1007/BF02872005.
[7]	H. Davenport and W. M. Schmidt, Dirichlet's theorem on diophantine approximation. Ⅱ, Acta Arithmetica, 16 (1969/70), 413-424. doi: 10.4064/aa-16-4-413-424.
[8]	W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, 129. Readings in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.
[9]	D. Kleinbock, G. Margulis and J. B. Wang, Metric diophantine approximation for systems of linear forms via dynamics, International Journal of Number Theory, 6 (2010), 1139-1168. doi: 10.1142/S1793042110003423.
[10]	D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and diophantine approximation on manifolds, Annals of Mathematics (2), 148 (1998), 339-360. doi: 10.2307/120997.
[11]	D. Kleinbock and B. Weiss, Dirichlet's theorem on diophantine approximation and homogeneous flows, Journal of Modern Dynamics, 2 (2008), 43-62. doi: 10.3934/jmd.2008.2.43.
[12]	S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159. doi: 10.1017/S0143385700008282.
[13]	M. Ratner, On raghunathan's measure conjecture, Annals of Mathematics (2), 134 (1991), 545-607. doi: 10.2307/2944357.
[14]	M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8.
[15]	N. A. Shah, Expanding translates of curves and dirichlet-minkowski theorem on linear forms, Journal of the American Mathematical Society, 23 (2010), 563-589. doi: 10.1090/S0894-0347-09-00657-2.
[16]	N. A Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proceedings Mathematical Sciences, 106 (1996), 105-125. doi: 10.1007/BF02837164.
[17]	N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 281-304. doi: 10.1215/00127094-2009-027.
[18]	N. A. Shah, Equidistribution of expanding translates of curves and dirichlets theorem on diophantine approximation, Inventiones Mathematicae, 177 (2009), 509-532.
[19]	N. A. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 251-279. doi: 10.1215/00127094-2009-026.
[20]	L. Yang, Equidistribution of expanding curves in homogeneous spaces and diophantine approximation for square matrices, Proc. Amer. Math. Soc., 144 (2016), 5291-5308. doi: 10.1090/proc/13170.
[21]	L. Yang, Expanding curves in $\mathrm{T}^1(\mathbb{H}^n)$ under geodesic flow and equidistribution in homogeneous spaces, Israel J. Math., 216 (2016), 389-413. doi: 10.1007/s11856-016-1414-6.

show all references

References:

[1]	M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, On metric diophantine approximation in matrices and lie groups, C. R. Math. Acad. Sci. Paris, 353 (2015), 185-189. doi: 10.1016/j.crma.2014.12.007.
[2]	M. Aka, E. Breuillard, L. Rosenzweig and N. de Saxcé, Diophantine approximation on matrices and lie groups, Geom. Funct. Anal., 28 (2018), 1-57. doi: 10.1007/s00039-018-0436-0.
[3]	R. C. Baker, Dirichlet's theorem on Diophantine approximation, Math. Proc. Cambridge Philos. Soc., 83 (1978), 37-59. doi: 10.1017/S030500410005427X.
[4]	Y. Bugeaud, Approximation by algebraic integers and hausdorff dimension, Journal of the London Mathematical Society (2), 65 (2002), 547-559. doi: 10.1112/S0024610702003137.
[5]	S. G. Dani, On orbits of unipotent flows on homogeneous spaces, Ergodic Theory and Dynamical Systems, 4 (1984), 25-34. doi: 10.1017/S0143385700002248.
[6]	S. G. Dani and G. A. Margulis, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17. doi: 10.1007/BF02872005.
[7]	H. Davenport and W. M. Schmidt, Dirichlet's theorem on diophantine approximation. Ⅱ, Acta Arithmetica, 16 (1969/70), 413-424. doi: 10.4064/aa-16-4-413-424.
[8]	W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, 129. Readings in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9.
[9]	D. Kleinbock, G. Margulis and J. B. Wang, Metric diophantine approximation for systems of linear forms via dynamics, International Journal of Number Theory, 6 (2010), 1139-1168. doi: 10.1142/S1793042110003423.
[10]	D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and diophantine approximation on manifolds, Annals of Mathematics (2), 148 (1998), 339-360. doi: 10.2307/120997.
[11]	D. Kleinbock and B. Weiss, Dirichlet's theorem on diophantine approximation and homogeneous flows, Journal of Modern Dynamics, 2 (2008), 43-62. doi: 10.3934/jmd.2008.2.43.
[12]	S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159. doi: 10.1017/S0143385700008282.
[13]	M. Ratner, On raghunathan's measure conjecture, Annals of Mathematics (2), 134 (1991), 545-607. doi: 10.2307/2944357.
[14]	M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8.
[15]	N. A. Shah, Expanding translates of curves and dirichlet-minkowski theorem on linear forms, Journal of the American Mathematical Society, 23 (2010), 563-589. doi: 10.1090/S0894-0347-09-00657-2.
[16]	N. A Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proceedings Mathematical Sciences, 106 (1996), 105-125. doi: 10.1007/BF02837164.
[17]	N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 281-304. doi: 10.1215/00127094-2009-027.
[18]	N. A. Shah, Equidistribution of expanding translates of curves and dirichlets theorem on diophantine approximation, Inventiones Mathematicae, 177 (2009), 509-532.
[19]	N. A. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Mathematical Journal, 148 (2009), 251-279. doi: 10.1215/00127094-2009-026.
[20]	L. Yang, Equidistribution of expanding curves in homogeneous spaces and diophantine approximation for square matrices, Proc. Amer. Math. Soc., 144 (2016), 5291-5308. doi: 10.1090/proc/13170.
[21]	L. Yang, Expanding curves in $\mathrm{T}^1(\mathbb{H}^n)$ under geodesic flow and equidistribution in homogeneous spaces, Israel J. Math., 216 (2016), 389-413. doi: 10.1007/s11856-016-1414-6.

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