Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T13:14:47.965Z Has data issue: false hasContentIssue false
  • English
  • Français

On badly approximable numbers

Published online by Cambridge University Press:  26 February 2010

Wolfgang M. Schmidt
Wolfgang M. Schmidt
Affiliation:
University of Colorado, Boulder, U.S.A.
Get access

Extract

Write ‖θ‖ for the distance from the real number θ to the nearest integer. An n-tuple of real numbers (β1, …, βn) will be called badly approximable, if there is constant C > 0 such that

for all positive integers q. As is well known, a single number β is badly approximable if and only if the partial quotients in its continued fraction are bounded.

Type
Research Article
Information
Mathematika , Volume 12 , Issue 1 , June 1965 , pp. 10 - 20
Copyright
Copyright © University College London 1965

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. Cassels, J. W. S., “On a method of Marshall Hall”, Mathematika, 3 (1956), 109–110.Google Scholar
2. Davenport, H., “A note on Diophantine approximation”, Studies in mathematical analysis and related topics (Stanford University Press, 1962), 77–81.Google Scholar
3. Davenport, H., “A note on Diophantine approximation (II)”, Mathematika, 11 (1964), 50–58.CrossRefGoogle Scholar
4. Hall, M., “On the sum and product of continued fractions”, Annals of Math. (2), 48 (1947), 966–993.Google Scholar
5. Hodge, W. V. D. and Pedoe, D., Methods of algebraic geometry, Vol. 1 (Cambridge, 1947).Google Scholar