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On Mahler's compound bodies

Part of: Algebraic number theory: global fields Diophantine approximation, transcendental number theory Geometry of numbers

Published online by Cambridge University Press:  09 April 2009

Edward B. Burger
Edward B. Burger
Affiliation:
Williams College, Williamstown, Massachusetts 01267, USA
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Abstract

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Let 1 ≤ M ≤ N − 1 be integers and K be a convex, symmetric set in Euclidean N-space. Associated with K and M, Mahler identified the Mth compound body of K, (K)m, in Euclidean (MN)-space. The compound body (K)M is describable as the convex hull of a certain subset of the Grassmann manifold in Euclidean (MN)-space determined by K and M. The sets K and (K)M are related by a number of well-known inequalities due to Mahler.

Here we generalize this theory to the geometry of numbers over the adèle ring of a number field and prove theorems which compare an adelic set with its adelic compound body. In addition, we include a comparison of the adelic compound body with the adelic polar body and prove an adelic general transfer principle which has implications to Diophantine approximation over number fields.

MSC classification

Secondary: 11H06: Lattices and convex bodies 11R56: Adèle rings and groups 11J13: Simultaneous homogeneous approximation, linear forms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 55 , Issue 2 , October 1993 , pp. 183 - 215
Copyright
Copyright © Australian Mathematical Society 1993

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