Perfect Lattices in Euclidean Spaces
Lattices are discrete subgroups of maximal rank in a Euclidean space. To each such geometrical object, we can attach a canonical sphere packing which, assuming some regularity, has a density. The question of estimating the highest possible density of a sp
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Series editors
A. Chenciner S.S. Chern B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hormander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt
Editor-in-Chief
M. Berger
J. Coates
S.R.S. Varadhan
Springer-Verlag Berlin Heidelberg GmbH
Jacques Martinet
Perfect Lattices in Euclidean Spaces
Springer
Jacques Martinet Universite Bordeaux 1, Institut de Mathematiques cours de la Liberation 351 33405 Talence cedex France e-mail: [email protected]
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): Primary: llH31 Secondary: llH55, llH56, llH71
ISSN 0072-7830 ISBN 978-3-642-07921-4 ISBN 978-3-662-05167-2 (eBook) DOl 10.1007/978-3-662-05167-2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003. Softcover reprint of the hardcover 1st edition 2003 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN 10518958
41/3142/db - 5 4 3 2 1 0
Preface to the English Edition
This book discusses a beautiful and central problem in mathematics, which involves geometry, number theory, coding theory and group theory. The geometrical objects we consider are lattices, i.e. discrete subgroups of maximal rank in a Euclidean space. To such an object we attach its canonical sphere packing, namely the set of (non-overlapping) spheres centred at all points of the lattice whose common radius is half the minimal distance of two lattice points. Assuming some regularity conditions, a sphere packing has a density. The question of estimating the highest possible density of a sphere packing in a given dimension n is a fascinating and difficult problem: the answer is known only up to dimension 3, and the case of dimension 3 was settled very recently by Hales, who gave
