Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-04T09:13:35.888Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON BADLY APPROXIMABLE LINEAR FORMS

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  04 February 2011

MUMTAZ HUSSAIN
MUMTAZ HUSSAIN*
Affiliation:
Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria 3086, Australia (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we investigate the analogue of the classical badly approximable setup in which the distance to the nearest integer ‖⋅‖ is replaced by the sup norm |⋅|. In the case of one linear form we prove that the hybrid badly approximable set is of full Hausdorff dimension.

Keywords

Diophantine approximationsystems of linear formsHausdorff dimension

MSC classification

Secondary: 11J25: Diophantine inequalities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 262 - 266
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Dickinson, H., ‘The Hausdorff dimension of the systems of the simultaneously small linear forms’, Mathematika 40 (1993), 367–374.CrossRefGoogle Scholar
[2]Dickinson, D., Gramchev, T. and Yoshino, M., ‘Perturbations of vector fields on tori: resonant normal forms and diophantine phenomena’, Proc. Edinb. Math. Soc. 45 (2002), 731–759.CrossRefGoogle Scholar
[3]Dirichlet, L., ‘Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen’, S.-B. Preuss. Akad. Wiss. (1842), 93–95.Google Scholar
[4]Dodson, M. M. and Vickers, J. A. G., ‘Exceptional sets in Kolmogorov–Arnol’d–Moser theory’, J. Phys. A 19 (1986), 349–374.CrossRefGoogle Scholar
[5]Falconer, K. J., Fractal Geometry—Mathematical Foundations and Applications (Wiley, New York, 1990).Google Scholar
[6]Hussain, M. and Levesley, J., ‘The metrical theory of simultaneously small linear forms’, Preprint, arXiv:0910.3428.Google Scholar
[7]Jarník, V., ‘Über die simultanen Diophantischen Approximationen’, Math. Z. 33 (1931), 505–543.CrossRefGoogle Scholar
[8]Khintchine, A., ‘Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen’, Math. Ann. 92 (1924), 115–125.CrossRefGoogle Scholar
[9]Schmidt, W. M., ‘Badly approximable systems of linear forms’, J. Number Theory 1 (1969), 139–154.CrossRefGoogle Scholar