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Communications in Mathematics

Editor-in-Chief: Rossi, Olga

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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


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


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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.

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

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