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

top
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 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

How to cite

top

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

top
  1. Blichfeldt, H., 10.1090/S0002-9947-1914-1500976-6, Trans. Amer. Math. Soc., 15, 1914, 227-235, (1914) MR1500976DOI10.1090/S0002-9947-1914-1500976-6
  2. Cassels, J.W.S., 10.1112/jlms/s1-30.1.119, J. London Math. Soc., 30, 1955, 119-121, (1955) Zbl0065.28302MR0066432DOI10.1112/jlms/s1-30.1.119
  3. Davenport, H., Simultaneous Diophantine approximation, Proc. London Math. Soc., 3, 2, 1952, 406-416, (1952) Zbl0048.03204MR0054657
  4. Davenport, H., On a theorem of Furtwängler, J. London Math. Soc., 30, 1955, 185-195, (1955) Zbl0064.04501MR0067943
  5. Gruber, P.M., Lekkerkerker, C.G., Geometry of numbers, 1987, North Holland, Amsterdam, (1987) Zbl0611.10017MR0893813
  6. Mack, J.M., 10.1017/S1446788700020292, J. Austral. Math. Soc., Ser. A, 24, 1977, 266-285, (1977) Zbl0377.10020MR0472719DOI10.1017/S1446788700020292
  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 and , 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

NotesEmbed ?

top

You must be logged in to post comments.