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The set of badly approximable vectors is strongly C1 incompressible

Published online by Cambridge University Press:  23 May 2012

RYAN BRODERICK ,
LIOR FISHMAN ,
DMITRY KLEINBOCK ,
ASAF REICH  and
BARAK WEISS
RYAN BRODERICK
Affiliation:
Department of Mathematics, Brandeis University, Waltham MA 02454-9110, U.S.A. e-mail: ryan@math.northwestern.edu, lfishman@unt.edu, kleinboc@brandeis.edu, azmreich@brandeis.edu
LIOR FISHMAN
Affiliation:
Department of Mathematics, Brandeis University, Waltham MA 02454-9110, U.S.A. e-mail: ryan@math.northwestern.edu, lfishman@unt.edu, kleinboc@brandeis.edu, azmreich@brandeis.edu
DMITRY KLEINBOCK
Affiliation:
Department of Mathematics, Brandeis University, Waltham MA 02454-9110, U.S.A. e-mail: ryan@math.northwestern.edu, lfishman@unt.edu, kleinboc@brandeis.edu, azmreich@brandeis.edu
ASAF REICH
Affiliation:
Department of Mathematics, Brandeis University, Waltham MA 02454-9110, U.S.A. e-mail: ryan@math.northwestern.edu, lfishman@unt.edu, kleinboc@brandeis.edu, azmreich@brandeis.edu
BARAK WEISS
Affiliation:
Department of Mathematics, Ben Gurion University, Be'er Sheva, Israel84105 e-mail: barakw@math.bgu.ac.il
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Abstract

We prove that the countable intersection of C1-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in ℝd, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's (α,β)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 2 , September 2012 , pp. 319 - 339
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Aravinda, C. S.Bounded geodesics and Hausdorff dimension. Math. Proc. Camb. Phil. Soc. 116 (1994), no. 3, 505–511.CrossRefGoogle Scholar
[2]Broderick, R., Bugeaud, Y., Fishman, L., Kleinbock, D. and Weiss, B.Schmidt's game, fractals and numbers normal to no base. Math. Research Letters 17 (2010), 307–321.CrossRefGoogle Scholar
[3]Broderick, R., Fishman, L. and Kleinbock, D.Schmidt's game, fractals, and orbits of toral endomorphisms. Ergodic Theory Dynam. Systems 31 (2011), 1095–1107.CrossRefGoogle Scholar
[4]Dani, S. G.Bounded orbits of flows on homogeneous spaces. Comment. Math. Helv. 61 (1986), 636–660.CrossRefGoogle Scholar
[5]Dani, S. G.On orbits of endomorphisms of tori and the Schmidt game. Ergodic Theory Dynam. Systems 8 (1988), 523–529.CrossRefGoogle Scholar
[6]Dani, S. G. On badly approximable numbers, Schmidt games and bounded orbits of flows, in: Number theory and dynamical systems (York, 1987), 69–86. London Math. Soc. Lecture Note Ser., 134 (Cambridge University Press, 1989).CrossRefGoogle Scholar
[7]Durand, A.Sets with large intersection and ubiquity. Math. Proc. Camb. Phil. Soc. 144 (2008), no. 1, 119–144.CrossRefGoogle Scholar
[8]Einsiedler, M. and Tseng, J.Badly approximable systems of affine forms, fractals and Schmidt games. J. Reine Angew. Math. 660 (2011), 83–97.Google Scholar
[9]Falconer, K.Sets with large intersection properties. J. London Math. Soc (2). 49 (1994), 267–280.CrossRefGoogle Scholar
[10]Färm, D.Simultaneously non-dense orbits under different expanding maps. Dyn. Syst. 25 (2010), no. 4, 531–545.CrossRefGoogle 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. 209 (2010), 157–176.CrossRefGoogle Scholar
[12]Fishman, L.Schmidt's game on fractals Israel J. Math. 171 (2009), no. 1, 77–92.CrossRefGoogle Scholar
[13]Fishman, L.Schmidt's game, badly approximable matrices and fractals. J. Number Theory 129 (2009), no. 9, 2133–2153.CrossRefGoogle Scholar
[14]Furstenberg, H.Ergodic fractal measures and dimension conservation. Ergodic Theory Dynam. Systems 28 (2008), no. 2, 405–422.CrossRefGoogle Scholar
[15]Gehring, F. W.Topics in quasiconformal mappings. Lecture Notes in Math. 1508 (1992), 62–80.Google Scholar
[16]Hutchinson, J. E.Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), no. 5, 713–747.CrossRefGoogle Scholar
[17]Kleinbock, D., Lindenstrauss, E. and Weiss, B.On fractal measures and diophantine approximation. Selecta Math. 10 (2004), 479–523.CrossRefGoogle Scholar
[18]Kleinbock, D. and Weiss, B.Badly approximable vectors on fractals. Israel J. Math. 149 (2005), 137–170.CrossRefGoogle Scholar
[19]Kleinbock, D. and Weiss, B.Modified Schmidt games and Diophantine approximation with weights. Adv. Math. 223 (2010), 1276–1298.CrossRefGoogle Scholar
[20]Kleinbock, D. and Weiss, B. Modified Schmidt games and a conjecture of Margulis. Preprint, arXiv:1001.5017.Google Scholar
[21]Kristensen, S., Thorn, R. and Velani, S. L.Diophantine approximation and badly approximable sets. Adv. Math. 203 (2006), 132–169.CrossRefGoogle Scholar
[22]Kronecker, L.Zwei Sätse über Gleichungen mit ganzzahligen Coefcienten. J. Reine Angew. Math. 53 (1857), 173–175; see also Werke, Vol. 1, 103–108 (Chelsea Publishing Co., New York, 1968).Google Scholar
[23]McMullen, C.Winning sets, quasiconformal maps and Diophantine approximation. To appear in Geom. Funct. Anal. 20 (2010), 726–740.CrossRefGoogle Scholar
[24]Moshchevitin, N. G.A note on badly approximable affine forms and winning sets. Moscow Math. J 11 (2011), no. 1, 129–137.CrossRefGoogle 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. vol. 785 (Springer-Verlag, Berlin, 1980).Google Scholar
[29]Stratmann, B. and Urbanski, M.Diophantine extremality of the Pa terson measure. Math. Proc. Cam. Phil. Soc. 140 (2006), 297–304.CrossRefGoogle Scholar
[30]Tseng, J.Schmidt games and Markov partitions. Nonlinearity 22 (2009), no. 3, 525–543.CrossRefGoogle 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
[33]Weiss, B.Almost no points on a Cantor set are very well approximable. Proc. Roy. Soc. Lond. A 457 (2001), 949–952.CrossRefGoogle Scholar