# 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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topTong, Jing Cheng. "The best Diophantine approximation functions by continued fractions." Mathematica Bohemica 121.1 (1996): 89-94. <http://eudml.org/doc/247945>.

@article{Tong1996,

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
\begin\{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). \end\{align\}
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.},

author = {Tong, Jing Cheng},

journal = {Mathematica Bohemica},

keywords = {best diophantine approximation; continued fraction; diophantine approximation; continued fraction; best diophantine approximation},

language = {eng},

number = {1},

pages = {89-94},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {The best Diophantine approximation functions by continued fractions},

url = {http://eudml.org/doc/247945},

volume = {121},

year = {1996},

}

TY - JOUR

AU - Tong, Jing Cheng

TI - The best Diophantine approximation functions by continued fractions

JO - Mathematica Bohemica

PY - 1996

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 121

IS - 1

SP - 89

EP - 94

AB - 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
\begin{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). \end{align}
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.

LA - eng

KW - best diophantine approximation; continued fraction; diophantine approximation; continued fraction; best diophantine approximation

UR - http://eudml.org/doc/247945

ER -

## References

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