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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

References

[1]Aitken, A. C., Determinants and Matrices (Greenwood Press, Westport, 1938).Google Scholar
[2]Bombieri, E. and Vaaler, J., ‘On Siegel's lemma’, Invent. Math. 73 (1983), 11–32.CrossRefGoogle Scholar
[3]Burger, E. B., ‘Homogeneous Diophantine approximation in S-integers’, Pacific J. Math. 152 (1992), 211–253.CrossRefGoogle Scholar
[4]Fisher, E., ‘Über den Hadamardschen Determinantensatz’, Arch. Math. (Basel) 13 (1908), 32–40.Google Scholar
[5]John, F., ‘Extremum problems with inequalities as subsidiary conditions’, in: Studies and essays presented to R. Courant (Interscience, New York, 1948).Google Scholar
[6]Mahler, K., ‘On compound convex bodies I’, Proc. London Math. Soc. 5 (3) (1955), 358–379.CrossRefGoogle Scholar
[7]McFeat, R. B., Geometry of numbers in Adèle spaces, Dissertationes Math. 88 (Rozprawy Mat., 1971).Google Scholar
[8]Schlickewei, H. P., ‘The number of solutions occuring in the p-adic subspace theorem in diophantine approximation’, J. Reine Angew. Math. 406 (1990), 44–108.Google Scholar
[9]Schmidt, W. M., ‘Norm form equations’, Ann. of Math. 96 (1972), 526–551.CrossRefGoogle Scholar
[10]Schmidt, W. M., Diophantine Approximation, Lecture Notes in Math. 785 (Springer, Berlin, 1980).Google Scholar
[11]Schmidt, W. M., ‘The subspace theorem in diophantine approximation’, Compositio Math. 69 (1989), 121–173.Google Scholar
[12]Vaaler, J. D., ‘Small zeros of quadric forms over number fields’, Trans. Amer. Math. Soc. 302, (1987), 281–296.CrossRefGoogle Scholar
[13]Weil, A., Basic Number Theory (Springer, Berlin, 1974).CrossRefGoogle Scholar