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Badly approximable vectors, $C^{1}$ curves and number fields

Published online by Cambridge University Press:  11 February 2015

MANFRED EINSIEDLER ,
ANISH GHOSH  and
BEVERLY LYTLE
MANFRED EINSIEDLER
Affiliation:
ETH Zürich, Departement Mathematik, Rämistrasse 101 8092, Zürich, Switzerland email manfred.einsiedler@math.ethz.ch, beverly.lytle@math.ethz.ch
ANISH GHOSH
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India email ghosh@math.tifr.res.in
BEVERLY LYTLE
Affiliation:
ETH Zürich, Departement Mathematik, Rämistrasse 101 8092, Zürich, Switzerland email manfred.einsiedler@math.ethz.ch, beverly.lytle@math.ethz.ch
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Abstract

We show that the set of points on $C^{1}$ curves which are badly approximable by rationals in a number field form a winning set in the sense of Schmidt. As a consequence, we obtain a number field version of Schmidt’s conjecture in Diophantine approximation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 6 , September 2016 , pp. 1851 - 1864
Copyright
© Cambridge University Press, 2015 

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References

An, J.. Badziahin–Pollington–Velani’s theorem and Schmidt’s game. Bull. Lond. Math. Soc. 45(4) (2013), 721–733.Google Scholar
An, J.. Two dimensional badly approximable vectors and Schmidt’s game. Preprint, http://arxiv.org/abs/1204.3610.Google Scholar
Aravinda, C. S.. Bounded geodesics and Hausdorff dimension. Math. Proc. Cambridge Philos. Soc. 116(3) (1994), 505–511.Google Scholar
Badziahin, D., Pollington, A. and Velani, S.. On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture. Ann. of Math. (2) 174(3) (2011), 1837–1883.Google Scholar
Beresnevich, V.. Badly approximable points on manifolds. Preprint, http://arxiv.org/abs/1304.0571.Google Scholar
Broderick, R., Fishman, L., Kleinbock, D., Reich, A. and Weiss, B.. The set of badly approximable vectors is strongly C 1 incompressible. Math. Proc. Cambridge Philos. Soc. 153(2) (2012), 319–339.CrossRefGoogle Scholar
Burger, E.. Homogeneous Diophantine approximation in S-integers. Pacific J. Math. 152(2) (1992), 211–253.CrossRefGoogle Scholar
Dani, S. G.. Divergent trajectories of flows on homogeneous spaces and Diophantine approximation. J. Reine Angew. Math. 359 (1985), 55–89.Google Scholar
Dani, S. G.. On badly approximable numbers, Schmidt games and bounded orbits of flows. Number Theory and Dynamical Systems (London Mathematical Society Lecture Note Series, 134) . Eds. Dodson, M. M. and Vickers, J. A. G.. Cambridge University Press, Cambridge, 1989, pp. 69–86.CrossRefGoogle Scholar
Einsiedler, M., Katok, A. and Lindenstrauss, E.. Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann. of Math. (2) 164 (2006), 513–560.Google Scholar
Einsiedler, M. and Kleinbock, D.. Measure rigidity and p-adic Littlewood-type problems. Compositio Math. 143 (2007), 689–702.Google Scholar
Esdahl-Schou, R. and Kristensen, S.. On badly approximable complex numbers. Glasg. Math. J. 52(2) (2010), 349–355.Google Scholar
Falconer, K. J.. The Geometry of Fractal Sets (Cambridge Tracts in Mathematics, 85) . Cambridge University Press, Cambridge, 1986.Google Scholar
Hattori, T.. Some Diophantine approximation inequalities and products of hyperbolic spaces. J. Math. Soc. Japan 59(1) (2007), 239–264.Google Scholar
Jarnik, V.. Diophantischen approximationen und Hausdorffsches mass. Mat. Sb. 36 (1929), 371–382.Google Scholar
Kleinbock, D. and Tomanov, G.. Flows on $S$ -arithmetic homogeneous spaces and applications to metric Diophantine approximation, Max Planck Institute. Preprint, 2003.Google Scholar
Kleinbock, D. and Tomanov, G.. Flows on S-arithmetic homogenous spaces and application to metric Diophantine approximation. Comment. Math. Helv. 82 (2007), 519–581.CrossRefGoogle Scholar
Kleinbock, D. and Weiss, B.. Modified Schmidt games and Diophantine approximation with weights. Adv. Math. 223(4) (2010), 1276–1298.Google Scholar
Kleinbock, D. and Weiss, B.. Modified Schmidt games and a conjecture of Margulis. Preprint, 2010.Google Scholar
Kristensen, S., Thorn, R. and Velani, S.. Diophantine approximation and badly approximable sets. Adv. Math. 203 (2006), 132–169.CrossRefGoogle Scholar
Quême, R.. On Diophantine approximation by algebraic numbers of a given number field: a new generalization of Dirichlet approximation theorem. Journées Arithmétiques de Luminy: 17–21 Juillet 1989 (Astérisque, 198–200) . Ed. Lachaud, G.. Société Mathématique de France, Paris, 1992, pp. 273–283.Google Scholar
Schmidt, W. M.. On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 178–199.Google Scholar
Schmidt, W. M.. Diophantine Approximation (Lecture Notes in Mathematics, 785) . Springer, Berlin, 1980.Google Scholar
Schmidt, W. M.. Simultaneous approximation to algebraic numbers by elements of a number field. Monatsh. Math. 79 (1975), 55–66.Google Scholar