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Bounded orbits of certain diagonalizable flows on $ SL_{n}(R)/SL_{n}(Z)$


Authors: Lifan Guan and Weisheng Wu
Journal: Trans. Amer. Math. Soc. 370 (2018), 4661-4681
MSC (2010): Primary 11J04; Secondary 22E40, 28A78
DOI: https://doi.org/10.1090/tran/7082
Published electronically: December 20, 2017
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Abstract: We prove that the set of points that have bounded orbits under certain diagonalizable flows is a hyperplane absolute winning subset of $ SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z})$.


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  • [1] J. An, Badziahin-Pollington-Velani's theorem and Schmidt's game, The Bulletin of the London Mathematical Society 45 (2013), no. 4, 721-33.
  • [2] Jinpeng An, 2-dimensional badly approximable vectors and Schmidt's game, Duke Math. J. 165 (2016), no. 2, 267-284. MR 3457674, https://doi.org/10.1215/00127094-3165862
  • [3] Jinpeng An, Lifan Guan, and Dmitry Kleinbock, Bounded orbits of diagonalizable flows on $ {SL}_3(\mathbb{R})/{SL}_3(\mathbb{Z})$, Int. Math. Res. Not. IMRN 24 (2015), 13623-13652. MR 3436158, https://doi.org/10.1093/imrn/rnv120
  • [4] Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
  • [5] Dzmitry Badziahin, Andrew Pollington, and Sanju Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2) 174 (2011), no. 3, 1837-1883. MR 2846492, https://doi.org/10.4007/annals.2011.174.3.9
  • [6] Victor Beresnevich, Badly approximable points on manifolds, Invent. Math. 202 (2015), no. 3, 1199-1240. MR 3425389, https://doi.org/10.1007/s00222-015-0586-8
  • [7] Ryan Broderick, Lior Fishman, Dmitry Kleinbock, Asaf Reich, and Barak Weiss, The set of badly approximable vectors is strongly $ C^1$ incompressible, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 2, 319-339. MR 2981929, https://doi.org/10.1017/S0305004112000242
  • [8] J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. MR 1434478
  • [9] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math. 359 (1985), 55-89. MR 794799, https://doi.org/10.1515/crll.1985.359.55
  • [10] S. G. Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv. 61 (1986), no. 4, 636-660. MR 870710, https://doi.org/10.1007/BF02621936
  • [11] L. Fishman, D. S. Simmons and M. Urbański, Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (extended version), Memoirs of the American Mathematical Society, to appear.
  • [12] L. Guan and J. Yu, Badly approximable vectors in higher dimension, arXiv preprint arXiv:1509.08050, 2015.
  • [13] Dmitry Y. Kleinbock, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Math. J. 95 (1998), no. 1, 107-124. MR 1646538, https://doi.org/10.1215/S0012-7094-98-09503-5
  • [14] D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 141-172. MR 1359098, https://doi.org/10.1090/trans2/171/11
  • [15] Dmitry Kleinbock and Barak Weiss, Values of binary quadratic forms at integer points and Schmidt games, Recent trends in ergodic theory and dynamical systems, Contemp. Math., vol. 631, Amer. Math. Soc., Providence, RI, 2015, pp. 77-92. MR 3330339, https://doi.org/10.1090/conm/631/12597
  • [16] Grigorii A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 193-215. MR 1159213
  • [17] Dave Witte Morris, Ratner's theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005. MR 2158954
  • [18] Calvin C. Moore, Ergodicity of flows on homogeneous spaces, Amer. J. Math. 88 (1966), 154-178. MR 0193188, https://doi.org/10.2307/2373052
  • [19] Erez Nesharim and David Simmons, $ {\bf Bad} (s,t)$ is hyperplane absolute winning, Acta Arith. 164 (2014), no. 2, 145-152. MR 3224831, https://doi.org/10.4064/aa164-2-4
  • [20] Marina Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), no. 1, 235-280. MR 1106945, https://doi.org/10.1215/S0012-7094-91-06311-8
  • [21] Wolfgang M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178-199. MR 0195595, https://doi.org/10.2307/1994619
  • [22] Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR 568710

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Additional Information

Lifan Guan
Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100871, People’s Republic of China
Address at time of publication: Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom
Email: lifan.guan@york.ac.uk

Weisheng Wu
Affiliation: Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, 100083, People’s Republic of China
Email: wuweisheng@math.pku.edu.cn

DOI: https://doi.org/10.1090/tran/7082
Received by editor(s): April 28, 2016
Received by editor(s) in revised form: September 20, 2016
Published electronically: December 20, 2017
Additional Notes: The research of the second author was supported by CPSF (#2015T80010)
Article copyright: © Copyright 2017 American Mathematical Society

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