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Communications in Mathematics

Editor-in-Chief: Rossi, Olga

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Mathematical Citation Quotient (MCQ) 2016: 0.28

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2336-1298
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On the critical determinants of certain star bodies

Werner Georg Nowak
Published Online: 2017-06-28 | DOI: https://doi.org/10.1515/cm-2017-0002

Abstract

In a classic paper [14], 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

|x1|(|x1|3 + |x2|3 + |x3|3 ≤ 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

|x1|(|x1|3 + |x22 + x32)3/2≤ 1.

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

References

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About the article

Received: 2016-08-29

Accepted: 2016-10-27

Published Online: 2017-06-28

Published in Print: 2017-06-27


Citation Information: Communications in Mathematics, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2017-0002.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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