The best diophantine approximation functions by continued fractions

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

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