Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Approximation by continued fractions


Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 45 (1974), 323-324
MSC: Primary 10F20
DOI: https://doi.org/10.1090/S0002-9939-1974-0349594-X
MathSciNet review: 0349594
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ x$ be a real irrational number whose continued fraction has infinitely many partial quotients not less than $ k$. A simple proof is given that the inequality $ \vert x - p/q\vert < {({k^2} + 4)^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}{q^{ - 2}}$ has infinitely many rational solutions $ p/q$. The constant $ {({k^2} + 4)^{ - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$ is best possible.


References [Enhancements On Off] (What's this?)

  • [1] J. H. E. Cohn, Hurwitz' theorem, Proc. Amer. Math. Soc. 38 (1973), 436. MR 0313195 (47:1750)
  • [2] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Oxford Univ. Press, London, 1960.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10F20

Retrieve articles in all journals with MSC: 10F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0349594-X
Keywords: Hurwitz' theorem, continued fraction, diophantine approximation
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society