Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Communications in Mathematics

Editor-in-Chief: Rossi, Olga

2 Issues per year


Mathematical Citation Quotient (MCQ) 2016: 0.28

Open Access
Online
ISSN
2336-1298
See all formats and pricing
More options …

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

Werner Georg Nowak
Published Online: 2018-01-11 | DOI: https://doi.org/10.1515/cm-2017-0012

Abstract

In the problem of (simultaneous) Diophantine approximation in ℝ3 (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body K2 : (y2 + z2)(x2 + y2 + z2) ≤ 1 play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant ∆ (Kc) of more general star bodies Kc : (y2 + z2)c/2(x2 + y2 + z2) ≤ 1 ; where c is any positive constant. These are obtained by inscribing into Kc either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of c.

MSC 2010: 11J13; 11H16

Keywords: Geometry of numbers; critical determinant; simultaneous Diophantine approximation

References

  • [1] J.V. Armitage: On a method of Mordell in the geometry of numbers. Mathematika 2 (2) (1955) 132-140.CrossrefGoogle Scholar

  • [2] H. Davenport, K. Mahler: Simultaneous Diophantine approximation. Duke Math. J. 13 (1946) 105-111.Google Scholar

  • [3] H. Davenport: On a theorem of Furtwängler. J. London Math.Soc. 30 (1955) 185-195. Google Scholar

  • [4] P.M. Gruber, C.G. Lekkerkerker: Geometry of numbers. North Holland, Amsterdam (1987).Google Scholar

  • [5] H. Minkowski: Dichteste gitterförmige Lagerung kongruenter Körper. Nachr. Kön. Ges. Wiss. Göttingen (1904) 311-355.Google Scholar

  • [6] W.G. Nowak: The critical determinant of the double paraboloid and Diophantine approximation in R3 and R4. Math. Pannonica 10 (1999) 111-122.Google Scholar

  • [7] W.G. Nowak: Diophantine approximation in Rs: On a method of Mordell and Armitage. In: 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.Google Scholar

  • [8] W.G. Nowak: Lower bounds for simultaneous Diophantine approximation constants. Comm. Math. 22 (1) (2014) 71-76.Google Scholar

  • [9] W.G. Nowak: Simultaneous Diophantine approximation: Searching for analogues of Hurwitz's theorem. In: T.M. Rassias and P.M. Pardalos (eds.), Essays in mathematics and its applications. Springer, Switzerland (2016) 181-197.Google Scholar

  • [10] W.G. Nowak: On the critical determinants of certain star bodies. Comm. Math. 25 (1) (2017) 5-11.Google Scholar

  • [11] K. Ollerenshaw: The critical lattices of a sphere. J. London Math. Soc. 23 (1949) 297-299.Google Scholar

  • [12] J.V. Whitworth: The critical lattices of the double cone. Proc. London Math. Soc. 2 (1) (1951) 422-443.CrossrefGoogle Scholar

About the article

Received: 2017-02-21

Accepted: 2017-07-06

Published Online: 2018-01-11

Published in Print: 2017-12-20


Citation Information: Communications in Mathematics, Volume 25, Issue 2, Pages 149–157, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2017-0012.

Export Citation

© 2018. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in