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

On the number of good rational approximations to algebraic numbers


Authors: Julia Mueller and W. M. Schmidt
Journal: Proc. Amer. Math. Soc. 106 (1989), 859-866
MSC: Primary 11J68; Secondary 11J17
MathSciNet review: 961415
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Abstract: We study rational approximations $ x/y$ to algebraic and, more generally, to real numbers $ \xi $. Given $ \delta > 0$, and writing $ L = \log (1 + \delta )$, the number of approximations with $ \vert\xi - (x/y)\vert < {y^{ - 2 - \delta }}$ is $ \leq {L^{ - 1}}\log \log H + {c_1}(\delta ,r)$ if $ \xi $ is algebraic of degree $ \leq r$ and of height $ H$ , and is $ \leq {L^{ - 1}}\log \log B + {c_2}(\delta )$ if $ \xi $ is real and we restrict to approximations with $ y \leq B$. It turns out that the dependency on $ H$ resp. $ B$ in these estimates is the best possible, i.e., that the summands $ {L^{ - 1}}\log \log H$ resp. $ {L^{ - 1}}\log \log B$ are optimal.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0961415-1
PII: S 0002-9939(1989)0961415-1
Keywords: Rational approximation to algebraic numbers
Article copyright: © Copyright 1989 American Mathematical Society