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).

[2]

V. Beresnevich, Badly approximable points on manifolds,, preprint, (2013).

[3]

A. Borel, Linear Algebraic Groups,, Second edition, (1991). doi: 10.1007/978-1-4612-0941-6.

[4]

A. Borel, Introduction aux Groupes Arithmétiques,, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1341).

[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. doi: 10.4007/annals.2011.174.3.9.

[6]

D. Badziahin and S. Velani, Badly approximable points on planar curves and a problem of Davenport,, preprint, (2013).

[7]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces,, Invent. Math., 47 (1978), 101. doi: 10.1007/BF01578067.

[8]

_______, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55.

[9]

_______, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936.

[10]

_______, On badly approximable numbers, Schmidt games and bounded orbits of flows,, in Number Theory and Dynamical Systems (York, (1987), 69. doi: 10.1017/CBO9780511661983.006.

[11]

D. Dolgopyat, Bounded orbits of Anosov flows,, Duke Math. J., 87 (1997), 87. doi: 10.1215/S0012-7094-97-08704-4.

[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. doi: 10.4007/annals.2006.164.513.

[13]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces,, in Homogeneous Flows, (2010).

[14]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, Second edition, (2003). doi: 10.1002/0470013850.

[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. doi: 10.2307/1970838.

[16]

D. Kleinbock, Nondense orbits of flows on homogeneous spaces,, Ergodic Theory Dynam. Systems, 18 (1998), 373. doi: 10.1017/S0143385798100408.

[17]

________, Flows on homogeneous spaces and Diophantine properties of matrices,, Duke Math. J., 95 (1998), 107. doi: 10.1215/S0012-7094-98-09503-5.

[18]

D. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sinaĭ's Moscow Seminar on Dynamical Systems, (1996), 141.

[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, (2002), 813. doi: 10.1016/S1874-575X(02)80013-3.

[20]

D. Kleinbock and B. Weiss, Badly approximable vectors on fractals. Probability in mathematics,, Israel J. Math., 149 (2005), 137. doi: 10.1007/BF02772538.

[21]

_______, Dirichlet's theorem on diophantine approximation and homogeneous flows,, J. Mod. Dyn., 4 (2008), 43.

[22]

_______, Modified Schmidt games and diophantine approximation with weights,, Adv. Math., 223 (2010), 1276. doi: 10.1016/j.aim.2009.09.018.

[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, (1990), 193.

[24]

D. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005).

[25]

E. Nehsarim, Badly approximable vectors on a vertical Cantor set,, preprint, (2012).

[26]

Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications,, Chicago Lectures in Mathematics, (1997).

[27]

A. Pollington and S. Velani, On simultaneously badly approximable numbers,, J. London Math. Soc. (2), 66 (2002), 29. doi: 10.1112/S0024610702003265.

[28]

M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972).

[29]

M. Ratner, Interactions between ergodic theory, Lie groups, and number theory,, in Proceedings of the International Congress of Mathematicians, (1994), 157.

[30]

W. M. Schmidt, On badly approximable numbers and certain games,, Trans. Amer. Math. Soc., 123 (1966), 178. doi: 10.1090/S0002-9947-1966-0195595-4.

[31]

_______, Badly approximable systems of linear forms,, J. Number Theory, 1 (1969), 139. doi: 10.1016/0022-314X(69)90032-8.

[32]

_______, Open problems in Diophantine approximation,, in Diophantine Approximations and Transcendental Numbers (Luminy, 31 (1982), 271.

[33]

A. N. Starkov, The structure of orbits of homogeneous flows and the Raghunathan conjecture,, (Russian) Uspekhi Mat. Nauk, 45 (1990), 219. doi: 10.1070/RM1990v045n02ABEH002338.

[34]

G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces,, Duke Math. J., 119 (2003), 367. doi: 10.1215/S0012-7094-03-11926-2.

[35]

H. Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume,, (German) J. Reine Angew. Math., 246 (1971), 46.

[36]

B. Weiss, Finite-dimensional representations and subgroup actions on homogeneous spaces,, Israel J. Math., 106 (1998), 189. doi: 10.1007/BF02773468.

[37]

R. J. Zimmer, Ergodic Theory and Semisimple Groups,, Monographs in Mathematics, (1984).

show all references

References:
[1]

J. An, Two dimensional badly approximable vectors and Schmidt's game,, preprint, (2012).

[2]

V. Beresnevich, Badly approximable points on manifolds,, preprint, (2013).

[3]

A. Borel, Linear Algebraic Groups,, Second edition, (1991). doi: 10.1007/978-1-4612-0941-6.

[4]

A. Borel, Introduction aux Groupes Arithmétiques,, (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, (1341).

[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. doi: 10.4007/annals.2011.174.3.9.

[6]

D. Badziahin and S. Velani, Badly approximable points on planar curves and a problem of Davenport,, preprint, (2013).

[7]

S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces,, Invent. Math., 47 (1978), 101. doi: 10.1007/BF01578067.

[8]

_______, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55.

[9]

_______, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936.

[10]

_______, On badly approximable numbers, Schmidt games and bounded orbits of flows,, in Number Theory and Dynamical Systems (York, (1987), 69. doi: 10.1017/CBO9780511661983.006.

[11]

D. Dolgopyat, Bounded orbits of Anosov flows,, Duke Math. J., 87 (1997), 87. doi: 10.1215/S0012-7094-97-08704-4.

[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. doi: 10.4007/annals.2006.164.513.

[13]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces,, in Homogeneous Flows, (2010).

[14]

K. Falconer, Fractal Geometry. Mathematical Foundations and Applications,, Second edition, (2003). doi: 10.1002/0470013850.

[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. doi: 10.2307/1970838.

[16]

D. Kleinbock, Nondense orbits of flows on homogeneous spaces,, Ergodic Theory Dynam. Systems, 18 (1998), 373. doi: 10.1017/S0143385798100408.

[17]

________, Flows on homogeneous spaces and Diophantine properties of matrices,, Duke Math. J., 95 (1998), 107. doi: 10.1215/S0012-7094-98-09503-5.

[18]

D. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sinaĭ's Moscow Seminar on Dynamical Systems, (1996), 141.

[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, (2002), 813. doi: 10.1016/S1874-575X(02)80013-3.

[20]

D. Kleinbock and B. Weiss, Badly approximable vectors on fractals. Probability in mathematics,, Israel J. Math., 149 (2005), 137. doi: 10.1007/BF02772538.

[21]

_______, Dirichlet's theorem on diophantine approximation and homogeneous flows,, J. Mod. Dyn., 4 (2008), 43.

[22]

_______, Modified Schmidt games and diophantine approximation with weights,, Adv. Math., 223 (2010), 1276. doi: 10.1016/j.aim.2009.09.018.

[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, (1990), 193.

[24]

D. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005).

[25]

E. Nehsarim, Badly approximable vectors on a vertical Cantor set,, preprint, (2012).

[26]

Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications,, Chicago Lectures in Mathematics, (1997).

[27]

A. Pollington and S. Velani, On simultaneously badly approximable numbers,, J. London Math. Soc. (2), 66 (2002), 29. doi: 10.1112/S0024610702003265.

[28]

M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972).

[29]

M. Ratner, Interactions between ergodic theory, Lie groups, and number theory,, in Proceedings of the International Congress of Mathematicians, (1994), 157.

[30]

W. M. Schmidt, On badly approximable numbers and certain games,, Trans. Amer. Math. Soc., 123 (1966), 178. doi: 10.1090/S0002-9947-1966-0195595-4.

[31]

_______, Badly approximable systems of linear forms,, J. Number Theory, 1 (1969), 139. doi: 10.1016/0022-314X(69)90032-8.

[32]

_______, Open problems in Diophantine approximation,, in Diophantine Approximations and Transcendental Numbers (Luminy, 31 (1982), 271.

[33]

A. N. Starkov, The structure of orbits of homogeneous flows and the Raghunathan conjecture,, (Russian) Uspekhi Mat. Nauk, 45 (1990), 219. doi: 10.1070/RM1990v045n02ABEH002338.

[34]

G. Tomanov and B. Weiss, Closed orbits for actions of maximal tori on homogeneous spaces,, Duke Math. J., 119 (2003), 367. doi: 10.1215/S0012-7094-03-11926-2.

[35]

H. Wegmann, Die Hausdorff-Dimension von kartesischen Produkten metrischer Räume,, (German) J. Reine Angew. Math., 246 (1971), 46.

[36]

B. Weiss, Finite-dimensional representations and subgroup actions on homogeneous spaces,, Israel J. Math., 106 (1998), 189. doi: 10.1007/BF02773468.

[37]

R. J. Zimmer, Ergodic Theory and Semisimple Groups,, Monographs in Mathematics, (1984).

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