# American Institute of Mathematical Sciences

July  2013, 7(3): 429-460. doi: 10.3934/jmd.2013.7.429

## Modified Schmidt games and a conjecture of Margulis

 1 Goldsmith 207, Brandeis University, Waltham, MA 02454-9110 2 Ben Gurion University, Be'er Sheva, Israel 84105

Received  December 2012 Published  December 2013

We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.
Citation: Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429
##### References:
 [1] J. An, Two dimensional badly approximable vectors and Schmidt's game, preprint, (2012), arXiv:1204.3610. Google Scholar [2] V. Beresnevich, Badly approximable points on manifolds, preprint, (2013), arXiv:1304.0571. Google Scholar [3] A. Borel, Linear Algebraic Groups, Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.  Google Scholar [4] A. Borel, Introduction aux Groupes Arithmétiques, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.  Google Scholar [5] D. Badziahin, A. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2), 174 (2011), 1837-1883. doi: 10.4007/annals.2011.174.3.9.  Google Scholar [6] D. Badziahin and S. Velani, Badly approximable points on planar curves and a problem of Davenport, preprint, (2013), arXiv:1301.4243. Google Scholar [7] S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067.  Google Scholar [8] _______, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55.  Google Scholar [9] _______, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660. doi: 10.1007/BF02621936.  Google Scholar [10] _______, On badly approximable numbers, Schmidt games and bounded orbits of flows, in Number Theory and Dynamical Systems (York, 1987), London Math. Soc. Lecture Note Ser., 134, Cambridge Univ. Press, Cambridge, 1989, 69-86. doi: 10.1017/CBO9780511661983.006.  Google Scholar [11] D. Dolgopyat, Bounded orbits of Anosov flows, Duke Math. J., 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4.  Google Scholar [12] M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. Math. (2), 164 (2006), 513-560. doi: 10.4007/annals.2006.164.513.  Google Scholar [13] M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010.  Google Scholar [14] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003. doi: 10.1002/0470013850.  Google Scholar [15] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326. doi: 10.2307/1970838.  Google Scholar [16] D. Kleinbock, Nondense orbits of flows on homogeneous spaces, Ergodic Theory Dynam. Systems, 18 (1998), 373-396. doi: 10.1017/S0143385798100408.  Google Scholar [17] ________, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Math. J., 95 (1998), 107-124. doi: 10.1215/S0012-7094-98-09503-5.  Google Scholar [18] D. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.  Google Scholar [19] D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook on Dynamical Systems, Vol. 1A, North Holland, Amsterdam, 2002, 813-930. doi: 10.1016/S1874-575X(02)80013-3.  Google Scholar [20] D. Kleinbock and B. Weiss, Badly approximable vectors on fractals. Probability in mathematics, Israel J. Math., 149 (2005), 137-170. doi: 10.1007/BF02772538.  Google Scholar [21] _______, Dirichlet's theorem on diophantine approximation and homogeneous flows, J. Mod. Dyn., 4 (2008), 43-62. Google Scholar [22] _______, Modified Schmidt games and diophantine approximation with weights, Adv. Math., 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018.  Google Scholar [23] G. A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 193-215.  Google Scholar [24] D. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005.  Google Scholar [25] E. Nehsarim, Badly approximable vectors on a vertical Cantor set, preprint, (2012), arXiv:1204.0110. Google Scholar [26] Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.  Google Scholar [27] A. Pollington and S. Velani, On simultaneously badly approximable numbers, J. London Math. Soc. (2), 66 (2002), 29-40. doi: 10.1112/S0024610702003265.  Google Scholar [28] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar [29] M. Ratner, Interactions between ergodic theory, Lie groups, and number theory, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 157-182.  Google Scholar [30] W. M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178-199. doi: 10.1090/S0002-9947-1966-0195595-4.  Google Scholar [31] _______, Badly approximable systems of linear forms, J. Number Theory, 1 (1969), 139-154. doi: 10.1016/0022-314X(69)90032-8.  Google Scholar [32] _______, Open problems in Diophantine approximation, in Diophantine Approximations and Transcendental Numbers (Luminy, 1982), Progr. Math., 31, Birkhäuser, Boston, 1983, 271-287. Google Scholar [33] A. N. Starkov, The structure of orbits of homogeneous flows and the Raghunathan conjecture, (Russian) Uspekhi Mat. Nauk, 45 (1990), 219-220; translation in Russian Math. Surveys, 45 (1990), 227-228. doi: 10.1070/RM1990v045n02ABEH002338.  Google Scholar [34] G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces, Duke Math. J., 119 (2003), 367-392. doi: 10.1215/S0012-7094-03-11926-2.  Google Scholar [35] H. Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume, (German) J. Reine Angew. Math., 246 (1971), 46-75.  Google Scholar [36] B. Weiss, Finite-dimensional representations and subgroup actions on homogeneous spaces, Israel J. Math., 106 (1998), 189-207. doi: 10.1007/BF02773468.  Google Scholar [37] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.  Google Scholar

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##### References:
 [1] J. An, Two dimensional badly approximable vectors and Schmidt's game, preprint, (2012), arXiv:1204.3610. Google Scholar [2] V. Beresnevich, Badly approximable points on manifolds, preprint, (2013), arXiv:1304.0571. Google Scholar [3] A. Borel, Linear Algebraic Groups, Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.  Google Scholar [4] A. Borel, Introduction aux Groupes Arithmétiques, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.  Google Scholar [5] D. Badziahin, A. Pollington and S. Velani, On a problem in simultaneous Diophantine approximation: Schmidt's conjecture, Ann. of Math. (2), 174 (2011), 1837-1883. doi: 10.4007/annals.2011.174.3.9.  Google Scholar [6] D. Badziahin and S. Velani, Badly approximable points on planar curves and a problem of Davenport, preprint, (2013), arXiv:1301.4243. Google Scholar [7] S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., 47 (1978), 101-138. doi: 10.1007/BF01578067.  Google Scholar [8] _______, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55.  Google Scholar [9] _______, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660. doi: 10.1007/BF02621936.  Google Scholar [10] _______, On badly approximable numbers, Schmidt games and bounded orbits of flows, in Number Theory and Dynamical Systems (York, 1987), London Math. Soc. Lecture Note Ser., 134, Cambridge Univ. Press, Cambridge, 1989, 69-86. doi: 10.1017/CBO9780511661983.006.  Google Scholar [11] D. Dolgopyat, Bounded orbits of Anosov flows, Duke Math. J., 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4.  Google Scholar [12] M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. Math. (2), 164 (2006), 513-560. doi: 10.4007/annals.2006.164.513.  Google Scholar [13] M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010.  Google Scholar [14] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003. doi: 10.1002/0470013850.  Google Scholar [15] H. Garland and M. S. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups, Ann. of Math. (2), 92 (1970), 279-326. doi: 10.2307/1970838.  Google Scholar [16] D. Kleinbock, Nondense orbits of flows on homogeneous spaces, Ergodic Theory Dynam. Systems, 18 (1998), 373-396. doi: 10.1017/S0143385798100408.  Google Scholar [17] ________, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Math. J., 95 (1998), 107-124. doi: 10.1215/S0012-7094-98-09503-5.  Google Scholar [18] D. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.  Google Scholar [19] D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook on Dynamical Systems, Vol. 1A, North Holland, Amsterdam, 2002, 813-930. doi: 10.1016/S1874-575X(02)80013-3.  Google Scholar [20] D. Kleinbock and B. Weiss, Badly approximable vectors on fractals. Probability in mathematics, Israel J. Math., 149 (2005), 137-170. doi: 10.1007/BF02772538.  Google Scholar [21] _______, Dirichlet's theorem on diophantine approximation and homogeneous flows, J. Mod. Dyn., 4 (2008), 43-62. Google Scholar [22] _______, Modified Schmidt games and diophantine approximation with weights, Adv. Math., 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018.  Google Scholar [23] G. A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, in Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 193-215.  Google Scholar [24] D. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2005.  Google Scholar [25] E. Nehsarim, Badly approximable vectors on a vertical Cantor set, preprint, (2012), arXiv:1204.0110. Google Scholar [26] Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.  Google Scholar [27] A. Pollington and S. Velani, On simultaneously badly approximable numbers, J. London Math. Soc. (2), 66 (2002), 29-40. doi: 10.1112/S0024610702003265.  Google Scholar [28] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar [29] M. Ratner, Interactions between ergodic theory, Lie groups, and number theory, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 157-182.  Google Scholar [30] W. M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178-199. doi: 10.1090/S0002-9947-1966-0195595-4.  Google Scholar [31] _______, Badly approximable systems of linear forms, J. Number Theory, 1 (1969), 139-154. doi: 10.1016/0022-314X(69)90032-8.  Google Scholar [32] _______, Open problems in Diophantine approximation, in Diophantine Approximations and Transcendental Numbers (Luminy, 1982), Progr. Math., 31, Birkhäuser, Boston, 1983, 271-287. Google Scholar [33] A. N. Starkov, The structure of orbits of homogeneous flows and the Raghunathan conjecture, (Russian) Uspekhi Mat. Nauk, 45 (1990), 219-220; translation in Russian Math. Surveys, 45 (1990), 227-228. doi: 10.1070/RM1990v045n02ABEH002338.  Google Scholar [34] G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces, Duke Math. J., 119 (2003), 367-392. doi: 10.1215/S0012-7094-03-11926-2.  Google Scholar [35] H. Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume, (German) J. Reine Angew. Math., 246 (1971), 46-75.  Google Scholar [36] B. Weiss, Finite-dimensional representations and subgroup actions on homogeneous spaces, Israel J. Math., 106 (1998), 189-207. doi: 10.1007/BF02773468.  Google Scholar [37] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.  Google Scholar
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