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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
MathSciNet review: 0349594
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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.


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