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  • Author: Werner Georg Nowak x
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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 + |x2 2 + x3 2)3/2≤ 1.


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.