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

Werner Georg Nowak

Communications in Mathematics (2017)

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

Abstract

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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 : ( y 2 + z 2 ) ( x 2 + y 2 + z 2 ) ≤ 1 play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant Δ ( K c ) of more general star bodies K c : ( y 2 + z 2 ) c / 2 ( x 2 + y 2 + z 2 ) ≤ 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

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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

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  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
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