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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