| Title:
|
On the critical determinants of certain star bodies (English) |
| Author:
|
Nowak, Werner Georg |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
25 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
5-11 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In a classic paper, W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt's Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body $$ \lvert x_1\rvert ({\lvert x_1\rvert^3+\lvert x_2\rvert^3+\lvert x_3\rvert^3})\le 1\,.$$ In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved and extended to the star body $$ \lvert x_1\rvert (\lvert x_1\rvert^3+(x_2^2+x_3^2)^{3/2})\le 1\,. $$ (English) |
| Keyword:
|
Geometry of numbers |
| Keyword:
|
Diophantine approximation |
| Keyword:
|
approximation constants |
| Keyword:
|
critical determinant |
| MSC:
|
11H16 |
| MSC:
|
11J13 |
| idZBL:
|
Zbl 06888084 |
| idMR:
|
MR3667072 |
| . |
| Date available:
|
2017-09-01T12:10:14Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146840 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
[11] Nowak, W.G.: The critical determinant of the double paraboloid and Diophantine approximation in $\mathbb{R}^3$ and $\mathbb{R}^4$.Math. Pannonica, 10, 1999, 111-122, MR 1678107 |
| Reference:
|
[12] Nowak, W.G.: Simultaneous Diophantine approximation: Searching for analogues of Hurwitz's theorem.T.M. Rassias and P.M. Pardalos (eds.), Essays in mathematics and its applications, 2016, 181-197, Springer, Switzerland, MR 3526920 |
| Reference:
|
[13] Spohn, W.G.: Midpoint regions and simultaneous Diophantine approximation.Dissertation, Ann Arbor, Michigan, University Microfilms, Inc., Order No. 62-4343, (1962). MR 2613496 |
| Reference:
|
[14] Spohn, W.G.: Blichfeldt's theorem and simultaneous Diophantine approximation.Amer. J. Math., 90, 1968, 885-894, MR 0231794, 10.2307/2373489 |
| . |
