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

 

The conjugate property for Diophantine approximation of continued fractions


Author: Jing Cheng Tong
Journal: Proc. Amer. Math. Soc. 105 (1989), 535-539
MSC: Primary 11J04; Secondary 11J70, 11J72
MathSciNet review: 937852
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Abstract: Let $ \xi $ be an irrational number with simple continued fraction expansion $ \xi = [{a_0};{a_1}, \ldots ,{a_i}, \ldots ]$, and $ {p_i}/{q_i}$ be its $ i$th convergent. In this paper we first prove the duality of some inequalities, and then prove the following conjugate properties for symmetric and asymmetric Diophantine approximations.

(i) Among any three consecutive convergents $ {p_i}/{q_i}(i = n - 1,n,n + 1)$, at least one satisfies

$\displaystyle \xi - {p_i}/{q_i}\vert < 1/\left( {\sqrt {a_{^{n + 1}}^2 + 4q_i^2} } \right),$

and at least one does not satisfy this inequality.

(ii) Let $ \tau $ be a positive real number. Among any four consecutive convergents $ {p_i}/{q_i}(i = n - 1,n,n + 1,n + 2)$, at least one satisfies

$\displaystyle - 1/\left( {\sqrt {c_{^n}^2 + 4\tau q_i^2} } \right) < \xi - {p_i}/{q_i} < \tau /\left( {\sqrt {c_n^2 + 4\tau q_i^2} } \right),$

and at least one does not satisfy this inequality, where $ {c_n} = {a_{n + 1}}$ if $ n$ is odd, $ {c_n} = {a_{n + 2}}$ if $ n$ is even.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0937852-8
PII: S 0002-9939(1989)0937852-8
Article copyright: © Copyright 1989 American Mathematical Society