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Keywords:
Geometry of numbers; Diophantine approximation; approximation constants; critical determinant
Geometry of numbers; Diophantine approximation; approximation constants; critical determinant
Summary:
In a classic paper, 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 $$ \lvert x_1\rvert ({\lvert x_1\rvert^3+\lvert x_2\rvert^3+\lvert x_3\rvert^3})\le 1\,.$$ 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 $$ \lvert x_1\rvert (\lvert x_1\rvert^3+(x_2^2+x_3^2)^{3/2})\le 1\,. $$
References:
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