On the critical determinants of certain star bodies

Werner Georg Nowak

Communications in Mathematics (2017)

  • Volume: 25, Issue: 1, page 5-11
  • ISSN: 1804-1388

Abstract

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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 | x 1 | ( | x 1 | 3 + | x 2 | 3 + | x 3 | 3 ) ≤ 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 | x 1 | ( | x 1 | 3 + ( x 2 2 + x 3 2 ) 3 / 2 ) ≤ 1 .

How to cite

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Nowak, Werner Georg. "On the critical determinants of certain star bodies." Communications in Mathematics 25.1 (2017): 5-11. <http://eudml.org/doc/294144>.

@article{Nowak2017,
abstract = {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 \[ \vert x\_1\vert (\{\vert x\_1\vert ^3+\vert x\_2\vert ^3+\vert x\_3\vert ^3\})\le 1\,.\] 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 \[ \vert x\_1\vert (\vert x\_1\vert ^3+(x\_2^2+x\_3^2)^\{3/2\})\le 1\,. \]},
author = {Nowak, Werner Georg},
journal = {Communications in Mathematics},
keywords = {Geometry of numbers; Diophantine approximation; approximation constants; critical determinant},
language = {eng},
number = {1},
pages = {5-11},
publisher = {University of Ostrava},
title = {On the critical determinants of certain star bodies},
url = {http://eudml.org/doc/294144},
volume = {25},
year = {2017},
}

TY - JOUR
AU - Nowak, Werner Georg
TI - On the critical determinants of certain star bodies
JO - Communications in Mathematics
PY - 2017
PB - University of Ostrava
VL - 25
IS - 1
SP - 5
EP - 11
AB - 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 \[ \vert x_1\vert ({\vert x_1\vert ^3+\vert x_2\vert ^3+\vert x_3\vert ^3})\le 1\,.\] 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 \[ \vert x_1\vert (\vert x_1\vert ^3+(x_2^2+x_3^2)^{3/2})\le 1\,. \]
LA - eng
KW - Geometry of numbers; Diophantine approximation; approximation constants; critical determinant
UR - http://eudml.org/doc/294144
ER -

References

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  7. Mullender, P., Lattice points in non-convex regions I., Proc. Kon. Ned. Akad. Wet., 51, 1948, 874-884, (1948) Zbl0031.11301MR0027301
  8. Mullender, P., 10.2307/1969477, Ann. Math., 52, 1950, 417-426, (1950) Zbl0037.17102MR0037326DOI10.2307/1969477
  9. Niven, I., Zuckerman, H.S., Einführung in die Zahlentheorie, 1975, Bibliograph. Inst., Mannheim, (1975) MR0392779
  10. Nowak, W.G., 10.1007/BF01174811, Manuscr. math., 36, 1981, 33-46, (1981) Zbl0455.10020MR0637853DOI10.1007/BF01174811
  11. Nowak, W.G., The critical determinant of the double paraboloid and Diophantine approximation in ℝ 3 and ℝ 4 , Math. Pannonica, 10, 1999, 111-122, (1999) MR1678107
  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, (2016) MR3526920
  13. Spohn, W.G., Midpoint regions and simultaneous Diophantine approximation, Dissertation, Ann Arbor, Michigan, University Microfilms, Inc., Order No. 62-4343, (1962). MR2613496
  14. Spohn, W.G., 10.2307/2373489, Amer. J. Math., 90, 1968, 885-894, (1968) MR0231794DOI10.2307/2373489

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