Geroldinger A., Ruzsa I. Combinatorial number theory and additive group theory

Birkhäuser, 2009. — 325 p. — (Advanced Courses in Mathematics CRM Barcelona). — ISBN: 978-3-7643-8961-1.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 are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory.This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects the notes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.Foreword
I Additive Group Theory and Non-unique Factorizations
Alfred GeroldingerNotationBasic concepts of non-unique factorizations
Arithmetical invariants
Krullmonoids
Transfer principles
Main problems in factorization theoryThe Davenport constant and first precise arithmetical results
The Davenport constant
Group algebras
Arithmetical invariants againThe structure of sets of lengths
Unions of sets of lengths
Almost arithmetical multiprogressions
The characterization problemAddition theorems and direct zero-sum problems
The theorems of Kneser and of Kemperman-Scherk
On the Erd˝os–Ginzburg–Ziv constant s(G) and on some of its variantsInverse zero-sum problems and arithmetical consequences
Cyclic groups
Groups of higher rank
Arithmetical consequencesII Sumsets and StructureCardinality inequalitiesPl¨unnecke’s method
Magnification and disjoint paths
Layered product
The independent addition graph
Different summands
Pl¨unnecke’s inequality with a large subset
Sums and differences
Double and triple sums
A + B and A + B
On the non-commutative caseStructure of sets with few sumsTorsion groups
Freiman isomorphism and small models
Elements of Fourier analysis on groups
Bohr sets in sumsets
Some facts from the geometry of numbers
A generalized arithmetical progression in a Bohr set
Freiman’s theorem
Arithmetic progressions in sets with small sumsetLocation and sumsetsThe Cauchy–Davenport inequality
Kneser’s theorem
Sumsets and diameter, part
The impact function
Estimates for the impact function in one dimension
Multi-dimensional sets
Results using cardinality and dimension
The impact function and the hull volume
The impact volume
Hovanskii’s theoremDensity
Asymptotic and Schnirelmann density
Schirelmann’s inequality
Mann’s theorem
Schnirelmann’s theoremrevisited
Kneser’s theorem, density form
Adding a basis: Erd˝os’ theorem
Adding a basis: Pl¨unnecke’s theorem, density form
Adding the set of squares or primes
Essential componentsMeasure and topologyRaikov’s theorem and generalizations
The impact function
Meditation on convexity and dimension
Topologies on integers
The finest compactification
Banach density
The difference set topology
ExercisesIII Thematic seminars
A survey on additive and multiplicative decompositions of sumsets and of shifted sets, Christian ElsholtzPart I
Multiplicative decompositions of sumsets
Multiplicative decompositions of shifted sets
Some background from sieve methods
Proof of Theorem?
Part II
Sumsets
Countingmethods
A result from extremal graph theory
The case of primes
Chains of primes in arithmetic progressionsOn the detailed structure of sets with small additive property, Gregory A Freiman
Small doubling
Sums of different setsThe isoperimetric method,
Yahya O Hamidoune
Universal bounds of set products
The Frobenius problem
The α + β-theorems
Direct theorems
Inverse theorems
Torsion-free groups: The isoperimetric approach
The isoperimetric formalism
The structure of k-atomsAdditive structure of difference sets,
Norbert Hegyv´ariThe caseD(A)
An intermezzo; D(D(A)) is highly well structured
First difference sets and Bohr sets I
Raimi’s theorem; difference set of partitions
First difference sets and Bohr sets II; Følner’s theorem
Applications of Følner theoremThe polynomial method in additive combinatorics, Gyula K´arolyiApplications of the polynomial method to additive problems
Appendix: The Vandermonde matrixProblems in additive number theory, III,
Melvyn B Nathanson
What sets are sumsets?
Describing the structure of hA as h→∞
Representation functions
Sets with more sums than differences
Comparative theory of linear forms
The fundamental theorem of additive number theory
Thin asymptotic bases
Minimal asymptotic bases
Maximal asymptotic non-bases
Complementing sets of integers
The Caccetta–H¨aggkvist conjectureIncidences and the spectra of graphs, Jozsef SolymosiThe sum-product problem
The sum-product graph
The spectral bound
Three-term arithmetic progressions
The -AP graph
Sidon functions
Sidon functions
A bipartite incidence graph
The spectral bound
Incidence bounds on pseudolines
The incidence bound
Strongly regular graphsMulti-dimensional inverse additive problems, Yonutz V Stanchescu
Direct and inverse problems of additive and combinatorial number theory
The simplest inverse problem for sums of sets in several dimensions
Small doubling property on the plane
Planar sets with no three collinear points on a line
Exact structure results
Difference sets
Finite Abelian groups