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On the critical determinants of certain star bodies. (English) Zbl 1433.11085

Summary: In a classic paper [Am. J. Math. 90, 885–894 (1968; Zbl 0169.37801)], 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 \[ |x_1|({|x_1|^3+|x_2|^3+|x_3|^3})\leq1. \] 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 \[ |x_1|(|x_1|^3+(x_2^2+x_3^2)^{3/2})\leq1. \]

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11H16 Nonconvex bodies

Citations:

Zbl 0169.37801
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References:

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