On the critical determinants of certain star bodies

Abstract

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 + |x2 2 + x3 2)3/2≤ 1.