# The best Diophantine approximation functions by continued fractions

Mathematica Bohemica (1996)

• Volume: 121, Issue: 1, page 89-94
• ISSN: 0862-7959

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## Abstract

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Let $\xi =\left[{a}_{0};{a}_{1},{a}_{2},\cdots ,{a}_{i},\cdots \right]$ be an irrational number in simple continued fraction expansion, ${p}_{i}/{q}_{i}=\left[{a}_{0};{a}_{1},{a}_{2},\cdots ,{a}_{i}\right]$, ${M}_{i}={q}_{i}^{2}|\xi -{p}_{i}/{q}_{i}|$. In this note we find a function $G\left(R,r\right)$ such that $\begin{array}{ccc}& {M}_{n+1}G\left(R,r\right),\hfill & \hfill {M}_{n+1}>R\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}{M}_{n-1}>r\phantom{\rule{4.0pt}{0ex}}\text{imply}\phantom{\rule{4.0pt}{0ex}}{M}_{n} Together with a result the author obtained, this shows that to find two best approximation functions $\stackrel{˜}{H}\left(R,r\right)$ and $\stackrel{˜}{L}\left(R,r\right)$ is a well-posed problem. This problem has not been solved yet.

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