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Schmidt’s game, fractals, and orbits of toral endomorphisms

Published online by Cambridge University Press:  05 August 2010

RYAN BRODERICK ,
LIOR FISHMAN  and
DMITRY KLEINBOCK
RYAN BRODERICK
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02454-9110, USA (email: ryanb@brandeis.edu, lfishman@brandeis.edu, kleinboc@brandeis.edu)
LIOR FISHMAN
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02454-9110, USA (email: ryanb@brandeis.edu, lfishman@brandeis.edu, kleinboc@brandeis.edu)
DMITRY KLEINBOCK
Affiliation:
Department of Mathematics, Brandeis University, Waltham, MA 02454-9110, USA (email: ryanb@brandeis.edu, lfishman@brandeis.edu, kleinboc@brandeis.edu)
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Abstract

Given an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1095 - 1107
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Broderick, R., Bugeaud, Y., Fishman, L., Kleinbock, D. and Weiss, B.. Schmidt’s game, fractals, and numbers normal to no base. Math. Res. Lett. 17(2) (2010), 309–323.CrossRefGoogle Scholar
[2]Bugeaud, Y., Harrap, S., Kristensen, S. and Velani, S.. On shrinking targets for ℤm actions on tori. Mathematika, to appear.Google Scholar
[3]Bugeaud, Y. and Laurent, M.. On exponents of homogeneous and inhomogeneous Diophantine approximation. Moscow Math. J. 5(4) (2005), 747–766, 972.CrossRefGoogle Scholar
[4]Cassels, J. W. S.. An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge, UK, 1957.Google Scholar
[5]Dani, S. G.. On orbits of endomorphisms of tori and the Schmidt game. Ergod. Th. & Dynam. Sys. 8 (1988), 523–529.Google Scholar
[6]de Mathan, B.. Numbers contravening a condition in density modulo 1. Acta Math. Acad. Sci. Hungar. 36 (1980), 237–241.CrossRefGoogle Scholar
[7]Einsiedler, M. and Tseng, J.. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445.Google Scholar
[8]Esdahl-Schou, R. and Kristensen, S.. On badly approximable complex numbers. Glasg. Math. J. 52 (2010), 349–355.CrossRefGoogle Scholar
[9]Falconer, K.. Fractal Geometry. Mathematical Foundations and Applications. John Wiley, Hoboken, NJ, 2003.CrossRefGoogle Scholar
[10]Färm, D.. Simultaneously non-dense orbits under different expanding maps. Preprint, arXiv:0904.4365v1.Google Scholar
[11]Färm, D., Persson, T. and Schmeling, J.. Dimension of countable intersections of some sets arising in expansions in non-integer bases. Fund. Math., to appear.Google Scholar
[12]Fishman, L.. Schmidt’s game on fractals. Israel J. Math. 171(1) (2009), 77–92.CrossRefGoogle Scholar
[13]Fishman, L.. Schmidt’s game, badly approximable matrices and fractals. J. Number Theory 129(9) (2009), 2133–2153.CrossRefGoogle Scholar
[14]Kleinbock, D.. Nondense orbits of flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 18 (1998), 373–396.Google Scholar
[15]Kleinbock, D.. Badly approximable systems of affine forms. J. Number Theory 79(1) (1999), 83–102.CrossRefGoogle Scholar
[16]Kleinbock, D., Lindenstrauss, E. and Weiss, B.. On fractal measures and Diophantine approximation. Selecta Math. 10 (2004), 479–523.CrossRefGoogle Scholar
[17]Kleinbock, D. and Weiss, B.. Badly approximable vectors on fractals. Israel J. Math. 149 (2005), 137–170.CrossRefGoogle Scholar
[18]Kleinbock, D. and Weiss, B.. Modified Schmidt games and Diophantine approximation with weights. Adv. Math. 223(4) (2010), 1276–1298.CrossRefGoogle Scholar
[19]Kristensen, S., Thorn, R. and Velani, S. L.. Diophantine approximation and badly approximable sets. Adv. Math. 203 (2006), 132–169.CrossRefGoogle Scholar
[20]Kronecker, L.. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. J. Reine Angew. Math. 53 (1857), 173–175; see also L. Kronecker. Werke, Vol. 1. Chelsea Publishing Co., New York, 1968, pp. 103–108.Google Scholar
[21]McMullen, C.. Winning sets, quasiconformal maps and Diophantine approximation, Geom. Funct. Anal., to appear.Google Scholar
[22]Moshchevitin, N. G.. A note on badly approximable affine forms and winning sets. Preprint, arXiv:0812.3998v2.Google Scholar
[23]Pollington, A. D.. On nowhere dense ∂-sets. Groupe de travail d’analyse ultramétrique 10(2) (1982–1983), Exp. No. 22, 2 pp.Google Scholar
[24]Pollington, A. D.. On the density of the sequence {η kξ}. Illinois J. Math. 23(4) (1979), 511–515.Google Scholar
[25]Pollington, A. D. and Velani, S. L.. Metric Diophantine approximation and ‘absolutely friendly’ measures. Selecta Math. 11 (2005), 297–307.CrossRefGoogle Scholar
[26]Schmidt, W. M.. On badly approximable numbers and certain games. Trans. Amer. Math. Soc. 123 (1966), 27–50.CrossRefGoogle Scholar
[27]Schmidt, W. M.. Badly approximable systems of linear forms. J. Number Theory 1 (1969), 139–154.CrossRefGoogle Scholar
[28]Schmidt, W. M.. Diophantine Approximation (Lecture Notes in Mathematics, 785). Springer, Berlin, 1980.Google Scholar
[29]Stratmann, B. and Urbanski, M.. Diophantine extremality of the Patterson measure. Math. Proc. Cambridge Philos. Soc. 140 (2006), 297–304.CrossRefGoogle Scholar
[30]Tseng, J.. Schmidt games and Markov partitions. Nonlinearity 22(3) (2009), 525–543.Google Scholar
[31]Tseng, J.. Badly approximable affine forms and Schmidt games. J. Number Theory 129 (2009), 3020–3025.CrossRefGoogle Scholar
[32]Urbanski, M.. Diophantine approximation of self-conformal measures. J. Number Theory 110 (2005), 219–235.CrossRefGoogle Scholar