On the critical determinants of certain star bodies

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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.

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Journal Information

CiteScore 2017: 0.33

SCImago Journal Rank (SJR) 2017: 0.128
Source Normalized Impact per Paper (SNIP) 2017: 0.476

Mathematical Citation Quotient (MCQ) 2017: 0.43

Target Group

researchers in the fields of: algebraic structures, calculus of variations, combinatorics, control and optimization, cryptography, differential equations, differential geometry, fuzzy logic and fuzzy set theory, global analysis, mathematical physics and number theory


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