# Estimating the critical determinants of a class of three-dimensional star bodies

• Volume: 25, Issue: 2, page 149-157
• ISSN: 1804-1388

top

## Abstract

top
In the problem of (simultaneous) Diophantine approximation in ${ℝ}^{3}$ (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body ${K}_{2}:\phantom{\rule{1.0em}{0ex}}\left({y}^{2}+{z}^{2}\right)\left({x}^{2}+{y}^{2}+{z}^{2}\right)\le 1$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $\Delta \left({K}_{c}\right)$ of more general star bodies ${K}_{c}:\phantom{\rule{1.0em}{0ex}}{\left({y}^{2}+{z}^{2}\right)}^{c/2}\left({x}^{2}+{y}^{2}+{z}^{2}\right)\le 1,$ where $c$ is any positive constant. These are obtained by inscribing into ${K}_{c}$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$.

## How to cite

top

Nowak, Werner Georg. "Estimating the critical determinants of a class of three-dimensional star bodies." Communications in Mathematics 25.2 (2017): 149-157. <http://eudml.org/doc/294547>.

@article{Nowak2017,
abstract = {In the problem of (simultaneous) Diophantine approximation in $\mathbb \{R\}^3$ (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body $K\_2:\quad (y^2+z^2)(x^2+y^2+z^2)\le 1$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $\Delta (K_c)$ of more general star bodies $K\_c:\quad (y^2+z^2)^\{c/2\}(x^2+y^2+z^2)\le 1,$ where $c$ is any positive constant. These are obtained by inscribing into $K_c$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$.},
author = {Nowak, Werner Georg},
journal = {Communications in Mathematics},
keywords = {Geometry of numbers; critical determinant; simultaneous Diophantine approximation},
language = {eng},
number = {2},
pages = {149-157},
publisher = {University of Ostrava},
title = {Estimating the critical determinants of a class of three-dimensional star bodies},
url = {http://eudml.org/doc/294547},
volume = {25},
year = {2017},
}

TY - JOUR
AU - Nowak, Werner Georg
TI - Estimating the critical determinants of a class of three-dimensional star bodies
JO - Communications in Mathematics
PY - 2017
PB - University of Ostrava
VL - 25
IS - 2
SP - 149
EP - 157
AB - In the problem of (simultaneous) Diophantine approximation in $\mathbb {R}^3$ (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body $K_2:\quad (y^2+z^2)(x^2+y^2+z^2)\le 1$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $\Delta (K_c)$ of more general star bodies $K_c:\quad (y^2+z^2)^{c/2}(x^2+y^2+z^2)\le 1,$ where $c$ is any positive constant. These are obtained by inscribing into $K_c$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$.
LA - eng
KW - Geometry of numbers; critical determinant; simultaneous Diophantine approximation
UR - http://eudml.org/doc/294547
ER -

## References

top
1. Armitage, J.V., 10.1112/S0025579300000772, Mathematika, 2, 2, 1955, 132-140, (1955) Zbl0066.03602MR0077574DOI10.1112/S0025579300000772
2. Davenport, H., Mahler, K., 10.1215/S0012-7094-46-01311-7, Duke Math. J., 13, 1946, 105-111, (1946) Zbl0060.12000MR0016068DOI10.1215/S0012-7094-46-01311-7
3. Davenport, H., On a theorem of Furtwängler, J. London Math.Soc., 30, 1955, 185-195, (1955) Zbl0064.04501MR0067943
4. Gruber, P.M., Lekkerkerker, C.G., Geometry of numbers, 1987, North Holland, Amsterdam, (1987) Zbl0611.10017MR0893813
5. Minkowski, H., Dichteste gitterförmige Lagerung kongruenter Körper, Nachr. Kön. Ges. Wiss. Göttingen, 1904, 311-355, (1904)
6. 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
7. Nowak, W.G., Diophantine approximation in ${ℝ}^{s}$: On a method of Mordell and Armitage, Algebraic number theory and Diophantine analysis. Proceedings of the conference held in Graz, Austria, August 30 to September 5, 1998, W. de Gruyter, Berlin, 2000, 339-349, (2000) MR1770472
8. Nowak, W.G., Lower bounds for simultaneous Diophantine approximation constants, Comm. Math., 22, 1, 2014, 71-76, (2014) Zbl1368.11063MR3233728
9. 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
10. Nowak, W.G., 10.1515/cm-2017-0002, Comm. Math., 25, 1, 2017, 5-11, (2017) MR3667072DOI10.1515/cm-2017-0002
11. Ollerenshaw, K., The critical lattices of a sphere, J. London Math. Soc., 23, 1949, 297-299, (1949) Zbl0036.31103MR0028353
12. Whitworth, J.V., 10.1112/plms/s2-53.6.422, Proc. London Math. Soc., 2, 1, 1951, 422-443, (1951) Zbl0044.04302MR0042452DOI10.1112/plms/s2-53.6.422

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.