The critical determinant of the double paraboloid and simultaneous Diophantine approximation
by
Werner Georg Nowak
Universität für Bodenkultur, Wien

Already around the last turn of centuries Minkowski developed a method to evaluate the critical determinant of three-dimensional O-symmetric convex bodies. This can be applied to show that, for the double paraboloid

P:    x2+y2+|z| <= 1 ,

one has \Delta(P) = ¹/₂. Further, based on classic work by Mordell and Armitage, an inequality relating the critical determinants of certain special star bodies of unequal dimension is established. A combination of these ideas yields improved bounds for the simultaneous Diophantine approximation constants (in the sense of Hurwitz' theorem), with respect to the Euclidean norm, for dimension 3, 4, 5.

References

[1] W.G. Nowak, The critical determinant of the double paraboloid and Diophantine approximation in R³ and R⁴. Math. Pannonica 10 (1999), 111-122.

[2] W.G. Nowak, Diophantine approximation in R^s: On a method of Mordell and Armitage. Proc.Conf. Algebraic Number Theory and Diophantine Analysis held at Graz 1998, to appear.

Date received: April 20, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-43.