# 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.

## How to cite

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Tong, 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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1. F. Bagemihl, J. R. McLaughlin, Generalization of some classical theorems concerning triples of consecutive convergents to simple continued fractions, J. Reine Angew. Math. 221 (1966), 146-149. (1966) Zbl0135.11104MR0183999
2. E. Borel, Contribution a l'analyse arithmetique du continu, J. Math. Pures Appl. 9 (1903), 329-375. (1903)
3. M. Fujiwara, Bemerkung zur Theorie der Approximation der irrationalen Zahlen durch rationale Zahlen, Tôhoku Math. J. 14 (1918), 109-115. (1918)
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5. W. J. LeVeque, 10.1307/mmj/1028989860, Michigan Math. J. 2 (1953), 1-6. (1953) MR0062784DOI10.1307/mmj/1028989860
6. M. Miller, 10.1007/BF01899402, Arch. Math. (Basel) 6 (1955), 253-258. (1955) MR0069850DOI10.1007/BF01899402
7. M. B. Nathanson, 10.1090/S0002-9939-1974-0349594-X, Proc. Amer. Math. Soc. 45 (1974), 323-324. (1974) Zbl0293.10015MR0349594DOI10.1090/S0002-9939-1974-0349594-X
8. O. Perron, Die Lehre von den Kettenbruchen I, II, 3rd ed. Teubner, Leipzig, 1954. (1954) MR0064172
9. B. Segre, Lattice points in the infinite domains and asymmetric Diophantine approximation, Duke Math. J. 12 (1945), 337-365. (1945) MR0012096
10. J. Tong, 10.1090/S0002-9939-1989-0937852-8, Proc. Amer. Math. Soc. 105 (1989), 535-539. (1989) MR0937852DOI10.1090/S0002-9939-1989-0937852-8
11. J. Tong, 10.1016/0022-314X(90)90102-W, J. Number Theory 34 (1990), 53-57. (1990) Zbl0714.11036MR1054557DOI10.1016/0022-314X(90)90102-W
12. J. Tong, 10.1017/S1446788700034273, J. Austral. Math. Soc. Ser. A 51 (1991), 324-330. (1991) Zbl0739.11026MR1124558DOI10.1017/S1446788700034273

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