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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Good approximations and continued fractions


Authors: Cor Kraaikamp and Pierre Liardet
Journal: Proc. Amer. Math. Soc. 112 (1991), 303-309
MSC: Primary 11J70; Secondary 11J71
MathSciNet review: 1062392
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Abstract: Let $ {({q_n})_n}$ be the sequence of best approximation denominators of an irrational number $ \alpha $. The set of real numbers $ x$ for which $ {q_n}x \to 0$ $ (\bmod 1)$ is studied. It is shown that a number $ x$ belongs to $ \alpha \mathbb{Z}(\bmod {\text{1)}}$ if and only if a simple condition on the speed of the convergence related to an arithmetic property of $ \alpha $ is satisfied. This set is uncountable whenever $ \alpha $ has unbounded partial quotients.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1062392-7
PII: S 0002-9939(1991)1062392-7
Keywords: Continued fractions, diophantine approximations
Article copyright: © Copyright 1991 American Mathematical Society