Geodesic
The best Diophantine approximation functions by continued fractions
Mathematica Bohemica, Tome 121 (1996) no. 1, pp. 89-94.

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Let $\xi=[a_0;a_1,a_2,\dots,a_i,\dots]$ be an irrational number in simple continued fraction expansion, $p_i/q_i=[a_0;a_1,a_2,\dots,a_i]$, $M_i=q_i^2 |\xi-p_i/q_i|$. In this note we find a function $G(R,r)$ such that \align{n+1}\text{ and }M_{n-1}\text{ imply }M_n>G(R,r), {n+1}>R\text{ and }M_{n-1}>r\text{ imply }M_n(R,r). \endalign Together with a result the author obtained, this shows that to find two best approximation functions $\tilde H(R,r)$ and $\tilde L(R,r)$ is a well-posed problem. This problem has not been solved yet.
DOI : 10.21136/MB.1996.125943
Classification : 11A55, 11J04, 11J70
Keywords: best diophantine approximation; continued fraction; diophantine approximation 
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Tong, Jingcheng. The best Diophantine approximation functions by continued fractions. Mathematica Bohemica, Tome 121 (1996) no. 1, pp. 89-94. doi : 10.21136/MB.1996.125943. http://geodesic.mathdoc.fr/articles/10.21136/MB.1996.125943/

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