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ON SHRINKING TARGETS FOR ℤm ACTIONS ON TORI

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  16 April 2010

Yann Bugeaud ,
Stephen Harrap ,
Simon Kristensen  and
Sanju Velani
Yann Bugeaud
Affiliation:
Département de Mathématiques, Université de Strasbourg, 7, rue René Descartes, F-67084 Strasbourg cedex, France (email: [email protected])
Stephen Harrap
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K. (email: [email protected])
Simon Kristensen
Affiliation:
Department of Mathematical Sciences, Faculty of Science, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark (email: [email protected])
Sanju Velani
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K. (email: [email protected])
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Abstract

Let A be an n×m matrix with real entries. Consider the set BadA of x∈[0,1)n for which there exists a constant c(x)>0 such that for any q∈ℤm the distance between x and the point {Aq} is at least c(x)|q|−m/n. It is shown that the intersection of BadA with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.

MSC classification

Secondary: 11K60: Diophantine approximation 11J83: Metric theory 37E10: Maps of the circle 37E45: Rotation numbers and vectors
Type
Research Article
Information
Mathematika , Volume 56 , Issue 2 , July 2010 , pp. 193 - 202
Copyright
Copyright © University College London 2010

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References

[1]Beresnevich, V., Bernik, V., Dodson, M. and Velani, S., Classical metric diophantine approximation revisited. In Analytic Number Theory: Essays in Honour of Klaus Roth, (eds Chen, W., Gowers, T., Halberstam, H., Schmidt, W. M. and Vaughan, R. C.), Cambridge University Press (Cambridge, 2009), 38–61.Google Scholar
[2]Beresnevich, V. and Velani, S., A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992.Google Scholar
[3]Bugeaud, Y., A note on inhomogeneous Diophantine approximation. Glasg. Math. J. 45(1) (2003), 105–110.Google Scholar
[4]Bugeaud, Y. and Chevallier, N., On simultaneous inhomogeneous Diophantine approximation. Acta Arith. 123(2) (2006), 97–123.CrossRefGoogle Scholar
[5]Bugeaud, Y. and Laurent, M., On exponents of homogeneous and inhomogeneous Diophantine approximation. Mosc. Math. J. 5(4) (2005), 747–766, 972.Google Scholar
[6]Cassels, J. W. S., An Introduction to Diophantine Approximation (Cambridge Tracts in Mathematics and Mathematical Physics 45), Cambridge University Press (New York, 1957).Google Scholar
[7]de Mathan, B., Sur un problème de densité modulo 1. C. R. Acad. Sci. Paris Sér. A–B 287(5) (1978), A277–A279.Google Scholar
[8]de Mathan, B., Numbers contravening a condition in density modulo 1. Acta Math. Acad. Sci. Hungar. 36(3–4) (1980), 237–241.Google Scholar
[9]Fan, A. H. and Wu, J., A note on inhomogeneous Diophantine approximation with a general error function. Glasg. Math. J. 48(2) (2006), 187–191.Google Scholar
[10]Fayad, B., Mixing in the absence of the shrinking target property. Bull. Lond. Math. Soc. 38(5) (2006), 829–838.Google Scholar
[11]Hill, R. and Velani, S., Ergodic theory of shrinking targets. Invent. Math. 119 (1995), 175–198.Google Scholar
[12]Khinchin, A. Ya., Sur le problème de Tchebycheff. Izv. Akad. Nauk SSSR Ser. Mat. 10 (1946), 281–294.Google Scholar
[13]Kim, D. H., The shrinking target property of irrational rotations. Nonlinearity 20(7) (2007), 1637–1643.Google Scholar
[14]Kleinbock, D. and Weiss, B., Badly approximable vectors on fractals. Israel J. Math. 149 (2005), 137–170.Google Scholar
[15]Kristensen, S., Thorn, R. and Velani, S., Diophantine approximation and badly approximable sets. Adv. Math. 203(1) (2006), 132–169.Google Scholar
[16]Kurzweil, J., On the metric theory of inhomogeneous diophantine approximations. Studia Math. 15 (1955), 84–112.CrossRefGoogle Scholar
[17]Mattila, P., Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge Studies in Advanced Mathematics 44), Cambridge University Press (Cambridge, 1995).CrossRefGoogle Scholar
[18]Minkowski, H., Über die Annäherung an eine reelle Grösse durch rationale Zahlen. Math. Ann. 54 (1901), 91–124 (in German).Google Scholar
[19]Pollington, A. D., On the density of sequence {nkξ}. Illinois J. Math. 23(4) (1979), 511–515.Google Scholar
[20]Pollington, A. D. and Velani, S., Metric Diophantine approximation and “absolutely friendly” measures. Selecta Math. (N.S.) 11 (2005), 297–307.Google Scholar
[21]Trubetskoĭ, S. and Schmeling, J., Inhomogeneous Diophantine approximations and angular recurrence for billiards in polygons. Mat. Sb. 194(2) (2003), 129–144.Google Scholar
[22]Tseng, J., On circle rotations and the shrinking target properties. Discrete Contin. Dyn. Syst. 20(4) (2008), 1111–1122.CrossRefGoogle Scholar
[23]Tseng, J., Three remarks on shrinking target properties. Preprint, 2008, arXiv:0807.3298v1.Google Scholar
[24]Weyl, H., Über die Gleichverteilung von Zahlen mod Eins. Math. Ann. 77(3) (1916), 313–352.Google Scholar