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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: m.hussain@latrobe.edu.au)
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Abstract

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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