Combinatorial Number Theory and Additive Group Theory
This book collects the material delivered in the 2008 edition of the DocCourse in Combinatorics and Geometry which was devoted to the topic of additive combinatorics. The first two parts, which form the bulk of the volume, contain the two main advanced co
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Alfred Geroldinger Imre Z. Ruzsa
Combinatorial Number Theory and Additive Group Theory
Birkhäuser Basel · Boston · Berlin
Authors: Alfred Geroldinger Institute for Mathematics and Scientific Computing University of Graz Heinrichstrasse 36 8010 Graz, Austria e-mail: [email protected]
Imre Z. Ruzsa Alfréd Rényi Institute of Mathematics P.O. Box 127 1364 Budapest, Hungary e-mail: [email protected]
2000 Mathematical Subject Classification 11P70, 11B50, 11R27
Library of Congress Control Number: 2008941509
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 .
ISBN 978-3-7643-8961-1 Birkhäuser Verlag AG, Basel – Boston – Berlin 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-8961-1
e-ISBN 978-3-7643-8962-8
987654321
www.birkhauser.ch
Foreword This book collects the material delivered in the 2008 edition of the DocCourse in Combinatorics and Geometry which was devoted to the topic of Additive Combinatorics. The two first parts, which form the bulk of the volume, contain the two main advanced courses, Additive Group Theory and Non-unique Factorizations, by Alfred Geroldinger, and Sumsets and Structure, by Imre Z. Ruzsa. The first part focusses on the interplay between zero-sum problems, arising from the Erd˝ os–Ginzburg–Ziv theorem, and nonuniqueness of factorizations in monoids and integral domains. The second part deals with structural set addition. It aims at describing the structure of sets in a commutative group from the knowledge of some properties of its sumset. The third part of the volume collects some of the seminars which accompanied the main courses and covers several aspects of contemporary methods and problems in Additive Combinatorics: multiplicative properties of sumsets (Christian Elsholtz), a step further in the inverse 3k − 4-theorem (Gregory A. Freiman), the isoperimetric method (Yahya O. Hamidoune), new developments around Følner’s theorem (Norbert Hegyv´ ari), the polynomial method (Gyula K´ arolyi), a survey on open problems (Melvyn B. Nathanson), spectral techniques for the sum-product problem (Jozsef Solymosi), and multidimensional inverse problems (Yonutz V. Stanchescu). We are grateful to Itziar Bardaj´ı and Llu´ıs Vena for their careful proofreading of all its chapters. This edition of the DocCourse has been supported by the Spanish project i-Math and by the Centre de Recerca Matem`atica, to which we express our
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