Combinatorial Number Theory and Additive Group Theory

325 Pages·2009·2.64 MB·English

by  Alfred Geroldinger

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
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ISBN 978-3-7643-8961-1 Birkhäuser Verlag AG, Basel – Boston – Berlin
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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