Hostname: page-component-848d4c4894-jbqgn Total loading time: 0 Render date: 2024-07-07T21:10:37.285Z Has data issue: false hasContentIssue false
  • English
  • Français

Simultaneous Diophantine approximation

Published online by Cambridge University Press:  09 April 2009

J. M. Mack
J. M. Mack
Affiliation:
Department of Pure Mathematics, University of Sydney, N.S.W. 2006, Australia.
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.

Using a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — αr)2 < c, r(q — βr)2 < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals α and β, provided that c > 1/2.6394. This improves an earlier result of Davenport, who shows that the same conclusion holds if c > 1/46¼ = 1/2.6043 ….

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 24 , Issue 3 , November 1977 , pp. 266 - 285
Copyright
Copyright © Australian Mathematical Society 1977

References

Cassels, J. W. S. (1955), ‘Simultaneous diophantine approximation’, J. London Math. Soc. 30, 119–121.Google Scholar
Cassels, J. W. S. (1959), An introduction to the geometry of numbers (Springer, 1959).Google Scholar
Davenport, H. (1952), ‘Simultaneous diophantine approximation’, Proc. London Math. Soc. (3) 2, 406–416.Google Scholar
Davenport, H. (1955), ‘On a theorem of Furtwängler’, J. London Math. Soc. 30, 186–195.Google Scholar
Davenport, H. and Mahler, K. (1946), ‘Simultaneous diophantine approximation’, Duke Math. J. 13, 105–111.CrossRefGoogle Scholar
Mack, J. M. (1971), Simultaneous diophantine approximation, Ph.D. thesis, University of Sydney.Google Scholar
Mullender, P. (1950), ‘Simultaneous approximation’, Annals of Mathematics 52, 417–426.Google Scholar