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Approximation of real numbers with rational number sequences

Author: Risto Korhonen
Journal: Proc. Amer. Math. Soc. 137 (2009), 107-113
MSC (2000): Primary 11J68; Secondary 11J97
Published electronically: August 14, 2008
MathSciNet review: 2439431
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Abstract: Let $ \alpha\in\mathbb{R}$, and let $ C>\max\{1,\alpha\}$. It is shown that if $ \{p_n/q_n\}$ is a sequence formed out of all rational numbers $ p/q$ such that

$\displaystyle \left\vert\alpha -\frac{p}{q} \right\vert \leq \frac{1}{Cq^2}, $

where $ p\in\mathbb{Z}$ and $ q\in\mathbb{N}$ are relatively prime numbers, then either $ \{p_n/q_n\}$ has finitely many elements or

$\displaystyle \limsup_{n\to\infty}\frac{\log\log q_n}{\log n}\geq 1, $

where the points $ \{q_n\}_{n\in\mathbb{N}}$ are ordered by increasing modulus. This implies that the sequence of denominators $ \{q_n\}_{n\in\mathbb{N}}$ grows exponentially as a function of $ n$, and so the density of rational numbers which approximate $ \alpha$ well in the above sense is relatively low.

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Additional Information

Risto Korhonen
Affiliation: Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland
Address at time of publication: Department of Mathematics and Statistics, P.O. Box 68, FI-00014, University of Helsinki, Finland

Received by editor(s): October 22, 2007
Received by editor(s) in revised form: January 9, 2008
Published electronically: August 14, 2008
Additional Notes: The research reported in this paper was supported in part by the Academy of Finland grants #118314 and #210245.
Communicated by: Ken Ono
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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