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MATHEMATICA BOHEMICA, Vol. 121, No. 1, pp. 89-94, 1996
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The best diophantine approximation functions by continued fractions

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Jingcheng Tong

* Jingcheng Tong*, Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, U.S.A

**Abstract:** 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&M_{n+1}<R\text{ and }M_{n-1}<r\text{ imply }M_n>G(R,r),&M_{n+1}>R\text{ and }M_{n-1}>r\text{ imply }M_n<G(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.

**Keywords:** continued fraction, diophantine approximation

**Classification (MSC91):** 11J04, 11A55

**Full text of the article:**

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