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Combinatorial Number Theory and Additive Group Theory / by Manuel Castellet, Alfred Geroldinger, Imre Z. Ruzsa
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Advanced Courses in Mathematics - CRM Barcelona, Centre de Recerca Matemàtica | SpringerLink BücherPublisher: Basel : Birkhäuser Basel, 2009Description: Online-Ressource (digital)ISBN:- 9783764389628
- 1280384387
- 9781280384387
- 512.7
- 511.1 23
- 511.6
- 510
- QA241
Contents:
Summary: Additive Group Theory and Non-unique Factorizations -- Notation -- Basic concepts of non-unique factorizations -- The Davenport constant and first precise arithmetical results -- The structure of sets of lengths -- Addition theorems and direct zero-sum problems -- Inverse zero-sum problems and arithmetical consequences -- Sumsets and Structure -- Notation -- Cardinality inequalities -- Structure of sets with few sums -- Location and sumsets -- Density -- Measure and topology -- Exercises -- Thematic seminars -- A survey on additive and multiplicative decompositions of sumsets and of shifted sets -- On the detailed structure of sets with small additive property -- The isoperimetric method -- Additive structure of difference sets -- The polynomial method in additive combinatorics -- Problems in additive number theory, III -- Incidences and the spectra of graphs -- Multi-dimensional inverse additive problems.Summary: 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 courses, Additive Group Theory and Non-Unique Factorizations by Alfred Geroldinger, and Sumsets and Structure by Imre Z. Ruzsa. The first part centers on the interaction between non-unique factorization theory and additive group theory. The main objective of factorization theory is a systematic treatment of phenomena related to the non-uniqueness of factorizations in monoids and domains. This part introduces basic concepts of factorization theory such as sets of lengths, and outlines the translation of arithmetical questions in Krull monoids into combinatorial questions on zero-sum sequences over the class group. Using methods from additive group theory such as the theorems of Kneser and of Kemperman-Scherk, classical zero-sum constants are studied, including the Davenport constant and the Erdös-Ginzburg-Ziv constant. Finally these results are applied again to the starting arithmetical problems. The second part is a course on the basics of combinatorial number theory (or additive combinatorics): cardinality inequalities (Plünnecke’s graph theoretical method), Freiman’s theorem on the structure of sets with a small sumset, inequalities for the Schnirelmann and asymptotic density of sumsets, analogous results for the measure of sumsets of reals, the connection with the Bohr topology. The third part of the volume collects some of the seminars which accompanied the main courses. It contains contributions by C. Elsholtz, G. Freiman, Y. O. Hamidoune, N. Hegyvari, G. Karolyi, M. Nathanson, J. Solymosi and Y. Stanchescu.PPN: PPN: 1648263666Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Foreword""; ""Contents""; ""Part I Additive Group Theory and Non-unique Factorizations""; ""Introduction""; ""Notation""; ""Chapter 1 Basic concepts of non-unique factorizations""; ""1.1 Arithmetical invariants""; ""1.2 Krull monoids""; ""1.3 Transfer principles""; ""1.4 Main problems in factorization theory""; ""Chapter 2 The Davenport constant and first precise arithmetical results""; ""2.1 The Davenport constant""; ""2.2 Group algebras""; ""2.3 Arithmetical invariants again""; ""Chapter 3 The structure of sets of lengths""; ""3.1 Unions of sets of lengths""
""3.2 Almost arithmetical multiprogressions and the structure of sets of lengths""""3.3 The characterization problem""; ""Chapter 4 Addition theorems and direct zero- sum problems""; ""4.1 The theorems of Kneser and of Kemperman-Scherk""; ""4.2 On the Erd. os�Ginzburg�Ziv constant s(G) and on some of its variants""; ""Chapter 5 Inverse zero-sum problems and arithmetical consequences""; ""5.1 Cyclic groups""; ""5.2 Groups of higher rank""; ""5.3 Arithmetical consequences""; ""Bibliography""; ""Part II Sumsets and Structure""; ""Introduction""; ""Notation""
""Chapter 1 Cardinality inequalities""""1.1 Introduction""; ""1.2 Plunnecke�s method""; ""1.3 Magnification and disjoint paths""; ""1.4 Layered product""; ""1.5 The independent addition graph""; ""1.6 Different summands""; ""1.7 Plunnecke�s inequality with a large subset""; ""1.8 Sums and differences""; ""1.9 Double and triple sums""; ""1.10 A + B and A + 2B""; ""1.11 On the non-commutative case""; ""Chapter 2 Structure of sets with few sums""; ""2.1 Introduction""; ""2.2 Torsion groups""; ""2.3 Freiman isomorphism and small models""; ""2.4 Elements of Fourier analysis on groups""
""2.5 Bohr sets in sumsets""""2.6 Some facts from the geometry of numbers""; ""2.7 A generalized arithmetical progression in a Bohr set""; ""2.8 Freiman�s theorem""; ""2.9 Arithmetic progressions in sets with small sumset""; ""Chapter 3 Location and sumsets""; ""3.1 Introduction""; ""3.2 The Cauchy�Davenport inequality""; ""3.3 Kneser�s theorem""; ""3.4 Sumsets and diameter, part 1""; ""3.5 The impact function""; ""3.6 Estimates for the impact function in one dimension""; ""3.7 Multi-dimensional sets""; ""3.8 Results using cardinality and dimension""
""3.9 The impact function and the hull volume""""3.10 The impact volume""; ""3.11 Hovanskii�s theorem""; ""Chapter 4 Density""; ""4.1 Asymptotic and Schnirelmann density""; ""4.2 Schirelmann�s inequality""; ""4.3 Mann�s theorem""; ""4.4 Schnirelmann�s theorem revisited""; ""4.5 Kneser�s theorem, density form""; ""4.6 Adding a basis: Erd. os� theorem""; ""4.7 Adding a basis: Plunnecke�s theorem, density form""; ""4.8 Adding the set of squares or primes""; ""4.9 Essential components""; ""Chapter 5 Measure and topology""; ""5.1 Introduction""
""5.2 Raikov�s theorem and generalizations""
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