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Combinatorial Number Theory and Additive Group Theory PDF

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Advanced Courses in Mathematics CRM Barcelona Centre de Recerca Matemàtica Managing Editor: Manuel Castellet Alfred Geroldinger Imre Z. Ruzsa Combinatorial Number Theory and Additive Group Theory Birkhäuser Basel · Boston · Berlin Authors: Alfred Geroldinger Imre Z. Ruzsa Institute for Mathematics and Alfréd Rényi Institute of Mathematics Scientific Computing P.O. Box 127 University of Graz 1364 Budapest, Hungary Heinrichstrasse 36 e-mail: [email protected] 8010 Graz, Austria 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 <http://dnb.ddb.de>. 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 9 8 7 6 5 4 3 2 1 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 Combi- natorics. 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 Erdo˝s–Ginzburg–Ziv theorem, and nonuniqueness of factorizations in monoids and integral domains. The second partdeals 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. Thethirdpartofthevolumecollectssomeoftheseminarswhichaccompanied the main courses and covers several aspects of contemporary methods and prob- lems 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(NorbertHegyva´ri),thepolynomialmethod(GyulaKa´rolyi),asurveyon open problems (Melvyn B. Nathanson), spectral techniques for the sum-product problem (Jozsef Solymosi), and multidimensional inverse problems (Yonutz V. Stanchescu). WearegratefultoItziarBardaj´ıandLlu´ısVenafortheircarefulproofreading 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 grat- itude. We particularly want to thank the director, Prof. Joaquim Bruna, and the staffoftheCentredeRecercaMatema`ticafortheirexcellentjobinorganizingthis edition of the DocCourse. Barcelona,July 2008 Javier Cilleruelo, Marc Noy and Oriol Serra Coordinators of the DocCourse Contents Foreword v I Additive Group Theory and Non-unique Factorizations Alfred Geroldinger 1 Introduction 3 Notation 5 1 Basic concepts of non-unique factorizations 7 1.1 Arithmetical invariants . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Krull monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Transfer principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Main problems in factorization theory . . . . . . . . . . . . . . . . 20 2 The Davenport constant and first precise arithmetical results 21 2.1 The Davenport constant . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Arithmetical invariants again . . . . . . . . . . . . . . . . . . . . . 29 3 The structure of sets of lengths 35 3.1 Unions of sets of lengths . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Almost arithmetical multiprogressions . . . . . . . . . . . . . . . . 38 3.3 The characterizationproblem . . . . . . . . . . . . . . . . . . . . . 42 4 Addition theorems and direct zero-sum problems 45 4.1 The theorems of Kneser and of Kemperman-Scherk . . . . . . . . . 45 4.2 OntheErdo˝s–Ginzburg–Zivconstants(G)andonsomeofitsvariants 48 5 Inverse zero-sum problems and arithmetical consequences 57 5.1 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Groups of higher rank . . . . . . . . . . . . . . . . . . . . . . . . . 70 viii Contents 5.3 Arithmetical consequences . . . . . . . . . . . . . . . . . . . . . . . 75 Bibliography 79 II Sumsets and Structure Imre Z. Ruzsa 87 Introduction 89 Notation 91 1 Cardinality inequalities 93 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 1.2 Plu¨nnecke’s method . . . . . . . . . . . . . . . . . . . . . . . . . . 95 1.3 Magnification and disjoint paths . . . . . . . . . . . . . . . . . . . 97 1.4 Layered product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 1.5 The independent addition graph . . . . . . . . . . . . . . . . . . . 101 1.6 Different summands . . . . . . . . . . . . . . . . . . . . . . . . . . 102 1.7 Plu¨nnecke’s inequality with a large subset . . . . . . . . . . . . . . 104 1.8 Sums and differences . . . . . . . . . . . . . . . . . . . . . . . . . . 106 1.9 Double and triple sums . . . . . . . . . . . . . . . . . . . . . . . . 109 1.10 A+B and A+2B . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 1.11 On the non-commutative case . . . . . . . . . . . . . . . . . . . . . 114 2 Structure of sets with few sums 119 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 2.2 Torsion groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 2.3 Freiman isomorphism and small models . . . . . . . . . . . . . . . 125 2.4 Elements of Fourier analysis on groups . . . . . . . . . . . . . . . . 128 2.5 Bohr sets in sumsets . . . . . . . . . . . . . . . . . . . . . . . . . . 132 2.6 Some facts from the geometry of numbers . . . . . . . . . . . . . . 134 2.7 A generalized arithmetical progressionin a Bohr set . . . . . . . . 135 2.8 Freiman’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2.9 Arithmetic progressions in sets with small sumset . . . . . . . . . . 139 3 Location and sumsets 141 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.2 The Cauchy–Davenportinequality . . . . . . . . . . . . . . . . . . 142 3.3 Kneser’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.4 Sumsets and diameter, part 1 . . . . . . . . . . . . . . . . . . . . . 146 3.5 The impact function . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.6 Estimates for the impact function in one dimension . . . . . . . . . 149 3.7 Multi-dimensional sets . . . . . . . . . . . . . . . . . . . . . . . . . 152 3.8 Results using cardinality and dimension . . . . . . . . . . . . . . . 154 Contents ix 3.9 The impact function and the hull volume . . . . . . . . . . . . . . 157 3.10 The impact volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3.11 Hovanskii’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4 Density 167 4.1 Asymptotic and Schnirelmann density . . . . . . . . . . . . . . . . 167 4.2 Schirelmann’s inequality . . . . . . . . . . . . . . . . . . . . . . . . 169 4.3 Mann’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.4 Schnirelmann’s theorem revisited . . . . . . . . . . . . . . . . . . . 173 4.5 Kneser’s theorem, density form . . . . . . . . . . . . . . . . . . . . 177 4.6 Adding a basis: Erdo˝s’ theorem . . . . . . . . . . . . . . . . . . . . 177 4.7 Adding a basis: Plu¨nnecke’s theorem, density form . . . . . . . . . 179 4.8 Adding the set of squares or primes. . . . . . . . . . . . . . . . . . 182 4.9 Essential components. . . . . . . . . . . . . . . . . . . . . . . . . . 184 5 Measure and topology 185 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.2 Raikov’s theorem and generalizations . . . . . . . . . . . . . . . . . 185 5.3 The impact function . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.4 Meditation on convexity and dimension . . . . . . . . . . . . . . . 188 5.5 Topologies on integers . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.6 The finest compactification . . . . . . . . . . . . . . . . . . . . . . 193 5.7 Banach density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.8 The difference set topology . . . . . . . . . . . . . . . . . . . . . . 196 Exercises 199 Bibliography 207 III Thematic seminars 211 1 A survey on additive and multiplicative decompositions of sumsets and of shifted sets, Christian Elsholtz 213 1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 1.2 Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 1.2.1 Multiplicative decompositions of sumsets . . . . . . . . . . 214 1.2.2 Multiplicative decompositions of shifted sets. . . . . . . . . 216 1.2.3 Some background from sieve methods . . . . . . . . . . . . 217 1.2.4 Proof of Theorem ?? . . . . . . . . . . . . . . . . . . . . . . 219 1.3 Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 1.3.1 Sumsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 1.3.2 Counting methods . . . . . . . . . . . . . . . . . . . . . . . 222 1.3.3 A result from extremal graph theory . . . . . . . . . . . . . 223 x Contents 1.3.4 The case of primes . . . . . . . . . . . . . . . . . . . . . . . 225 1.3.5 Chains of primes in arithmetic progressions . . . . . . . . . 226 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 2 On the detailed structure of sets with small additive property, Gregory A. Freiman 233 2.1 Small doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2.2 Sums of different sets. . . . . . . . . . . . . . . . . . . . . . . . . . 237 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 3 The isoperimetric method, Yahya O. Hamidoune 241 3.1 Universal bounds of set products . . . . . . . . . . . . . . . . . . . 241 3.2 The Frobenius problem . . . . . . . . . . . . . . . . . . . . . . . . 243 3.3 The α+β-theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3.3.1 Direct theorems. . . . . . . . . . . . . . . . . . . . . . . . . 244 3.3.2 Inverse theorems . . . . . . . . . . . . . . . . . . . . . . . . 245 3.4 Torsion-free groups: The isoperimetric approach. . . . . . . . . . . 246 3.5 The isoperimetric formalism . . . . . . . . . . . . . . . . . . . . . . 248 3.6 The structure of k-atoms . . . . . . . . . . . . . . . . . . . . . . . 250 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4 Additive structure of difference sets, Norbert Hegyva´ri 253 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 4.2 The case D(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 4.3 An intermezzo; D(D(A)) is highly well structured . . . . . . . . . . 258 4.4 First difference sets and Bohr sets I. . . . . . . . . . . . . . . . . . 258 4.5 Raimi’s theorem; difference set of partitions . . . . . . . . . . . . . 261 4.6 First difference sets and Bohr sets II; Følner’s theorem . . . . . . . 263 4.7 Applications of Følner theorem . . . . . . . . . . . . . . . . . . . . 263 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 5 The polynomial method in additive combinatorics, Gyula Ka´rolyi 267 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.2 Applications of the polynomial method to additive problems. . . . 271 5.3 Appendix: The Vandermonde matrix . . . . . . . . . . . . . . . . 275 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6 Problems in additive number theory, III, Melvyn B. Nathanson 279 6.1 What sets are sumsets? . . . . . . . . . . . . . . . . . . . . . . . . 279 6.2 Describing the structure of hA as h→∞ . . . . . . . . . . . . . . 280 Contents xi 6.3 Representation functions . . . . . . . . . . . . . . . . . . . . . . . . 280 6.4 Sets with more sums than differences . . . . . . . . . . . . . . . . . 282 6.5 Comparative theory of linear forms . . . . . . . . . . . . . . . . . . 284 6.6 The fundamental theorem of additive number theory . . . . . . . . 286 6.7 Thin asymptotic bases . . . . . . . . . . . . . . . . . . . . . . . . . 287 6.8 Minimal asymptotic bases . . . . . . . . . . . . . . . . . . . . . . . 288 6.9 Maximal asymptotic non-bases . . . . . . . . . . . . . . . . . . . . 289 6.10 Complementing sets of integers . . . . . . . . . . . . . . . . . . . . 290 6.11 The Caccetta–H¨aggkvistconjecture . . . . . . . . . . . . . . . . . . 293 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 7 Incidences and the spectra of graphs, Jozsef Solymosi 299 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 7.2 The sum-product problem . . . . . . . . . . . . . . . . . . . . . . . 300 7.2.1 The sum-product graph . . . . . . . . . . . . . . . . . . . . 301 7.2.2 The spectral bound . . . . . . . . . . . . . . . . . . . . . . 303 7.3 Three-term arithmetic progressions . . . . . . . . . . . . . . . . . . 304 7.3.1 The 3-AP graph . . . . . . . . . . . . . . . . . . . . . . . . 304 7.4 Sidon functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 7.4.1 Sidon functions . . . . . . . . . . . . . . . . . . . . . . . . . 306 7.4.2 A bipartite incidence graph . . . . . . . . . . . . . . . . . . 308 7.4.3 The spectral bound . . . . . . . . . . . . . . . . . . . . . . 309 7.5 Incidence bounds on pseudolines . . . . . . . . . . . . . . . . . . . 310 7.5.1 The incidence bound . . . . . . . . . . . . . . . . . . . . . . 310 7.5.2 Strongly regular graphs . . . . . . . . . . . . . . . . . . . . 311 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 8 Multi-dimensional inverse additive problems, Yonutz V. Stanchescu 315 8.1 Direct and inverse problems of additive and combinatorial number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 8.2 The simplest inverse problem for sums of sets in several dimensions 316 8.3 Small doubling property on the plane . . . . . . . . . . . . . . . . 318 8.4 Planar sets with no three collinear points on a line . . . . . . . . . 321 8.5 Exact structure results . . . . . . . . . . . . . . . . . . . . . . . . . 323 8.6 Difference sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.7 Finite Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . 325 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

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Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture
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