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Keywords:
Geometry of numbers; critical determinant; simultaneous Diophantine approximation
Geometry of numbers; critical determinant; simultaneous Diophantine approximation
Summary:
In the problem of (simultaneous) Diophantine approximation in~$\mathbb{R}^3$ (in the spirit of Hurwitz's theorem), lower bounds for the critical determinant of the special three-dimensional body $$ K_2:\quad (y^2+z^2)(x^2+y^2+z^2)\le 1 $$ play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant $\Delta (K_c)$ of more general star bodies $$ K_c:\quad (y^2+z^2)^{c/2}(x^2+y^2+z^2)\le 1, $$ where $c$ is any positive constant. These are obtained by inscribing into $K_c$ either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of $c$.
References:
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