Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T14:29:47.106Z Has data issue: false hasContentIssue false
  • English
  • Français

Metric Diophantine approximation and Hausdorff dimension on manifolds

Published online by Cambridge University Press:  04 October 2011

M. M. Dodson ,
B. P. Rynne  and
J. A. G. Vickers
M. M. Dodson
Affiliation:
Department of Mathematics, University of York, York YO1 5DD
B. P. Rynne
Affiliation:
Faculty of Mathematical Sciences, University of Southampton, Southampton SO9 5NH
J. A. G. Vickers
Affiliation:
Faculty of Mathematical Sciences, University of Southampton, Southampton SO9 5NH
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we discuss homogeneous Diophantine approximation of points on smooth manifolds M in ℝk. We begin with a brief survey of the notation and results. For any x,y ∈ℝk, let

.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 547 - 558
Copyright
Copyright © Cambridge Philosophical Society 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Arnold, V. I.. Geometrical Methods in the Theory of Ordinary Differential Equations (translated by Vogtmann, K. and Weinstein, A.) (Springer-Verlag, 1983).CrossRefGoogle Scholar
[2] Baker, R. C.. Sprindzuk's theorem and Hausdorff dimension. Mathematika 23 (1976), 184–197.CrossRefGoogle Scholar
[3] Baker, R. C.. Dirichlet's theorem on diophantine approximation. Math. Proc. Camb. Philos. Soc. 83 (1978), 37–59.CrossRefGoogle Scholar
[4] Bernik, V. I.. Application of Hausdorff dimension in the theory of Diophantine approximation. Acta Arith. 42 (1983), 219–253.Google Scholar
[5] Besicovitch, A. S.. Sets of fractional dimension. IV. On rational approximation to real numbers. J. London Math. Soc. 9 (1934), 126–131.CrossRefGoogle Scholar
[6] Bovey, J. D. and Dodson, M. M.. The Hausdorff dimension of systems of linear forms. Acta Arith. 45 (1986), 337–358.CrossRefGoogle Scholar
[7] Cassels, J. W. S.. An Introduction to Diophantine Approximation (Cambridge University Press, 1957).Google Scholar
[8] Dodson, M. M. and Vickers, J. A. G.. Exceptional sets in Kolmogorov—Arnol'd-Moser theory. J. Phys. A19 (1986), 349–374.Google Scholar
[9] Dodson, M. M., Rynne, B. P. and Vickers, J. A. G.. Averaging methods in multi-frequency systems. Nonlinearity (to appear).Google Scholar
[10] Dodson, M. M., Rynne, B. P. and Vickers, J. A. G.. Diophantine approximation and a lower bound for Hausdorff dimension. (To appear.)Google Scholar
[11] Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[12] Jarnik, V.. Diophantische Approximationen und Hausdorffsches Mass. Mat. Sb. 36 (1929), 371–382.Google Scholar
[13] Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry, vol. 2 (Interscience, 1969).Google Scholar
[14] Kovalevskaya, I.. A geometric property of extremal surfaces. Math. Notes 23 (1978), 99–101.CrossRefGoogle Scholar
[15] Melnichuk, Y. V.. Diophantine approximation on curves and Hausdorff dimension. Dokl. Akad. Nauk Ukrain. SSR Ser. A9 (1978), 793–796.Google Scholar
[16] Schmidt, W. M.. Metrische Sätze über simultane Approximation abhängiger Grössen. Monatsh. Math. 68 (1964), 154–166.CrossRefGoogle Scholar
[17] Schmidt, W. M.. Diophantine Approximation. Lecture Notes in Math. vol. 785 (Springer-Verlag, 1980).Google Scholar
[18] Sprindzuk, V. G.. Mahler's Problem in Metric Number Theory. Transl. Math. Monographs, vol. 25 (American Mathematical Society, 1969).Google Scholar
[19] Sprindzuk, V. G.. Metric Theory of Diophantine Approximations (translated by Silverman, R. A.) (Wiley, 1979).Google Scholar