| Title:
|
The best Diophantine approximation functions by continued fractions (English) |
| Author:
|
Tong, Jingcheng |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
121 |
| Issue:
|
1 |
| Year:
|
1996 |
| Pages:
|
89-94 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
best diophantine approximation |
| Keyword:
|
continued fraction |
| Keyword:
|
diophantine approximation |
| MSC:
|
11A55 |
| MSC:
|
11J04 |
| MSC:
|
11J70 |
| idZBL:
|
Zbl 0863.11042 |
| idMR:
|
MR1388180 |
| DOI:
|
10.21136/MB.1996.125943 |
| . |
| Date available:
|
2009-09-24T21:16:07Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/125943 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[5] W. J. LeVeque: On asymmetric approximations.Michigan Math. J. 2 (1953), 1-6. MR 0062784, 10.1307/mmj/1028989860 |
| Reference:
|
[6] M. Miller: Über die Approximation reeller Zahlen durch die Näherungsbruche ihres regelmässigen Kettenbruches.Arch. Math. (Basel) 6 (1955), 253-258. MR 0069850, 10.1007/BF01899402 |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
[11] J. Tong: Diophantine approximation of a single irrational number.J. Number Theory 34 (1990), 53-57. Zbl 0714.11036, MR 1054557, 10.1016/0022-314X(90)90102-W |
| Reference:
|
[12] J. Tong: Diophantine approximation by continued fractions.J. Austral. Math. Soc. Ser. A 51 (1991), 324-330. Zbl 0739.11026, MR 1124558, 10.1017/S1446788700034273 |
| . |
