# Combinatorial Number Theory and Additive Group Theory

• Alfred Geroldinger
• Imre Z. Ruzsa
Book

Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

1. Front Matter
Pages i-xi
2. ### Additive Group Theory and Non-unique Factorizations

1. Front Matter
Pages 1-1
2. Pages 3-3
3. Pages 5-5
4. Pages 35-44
3. ### Sumsets and Structure

1. Front Matter
Pages 87-87
2. Pages 89-90
3. Pages 91-91
4. Pages 93-117
5. Pages 119-140
6. Pages 141-165
7. Pages 167-184
8. Pages 185-197
9. Pages 199-210
4. ### Thematic seminars

1. Front Matter
Pages 211-211
2. Christian Elsholtz
Pages 213-231

### Introduction

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.

### Keywords

Graph Graph theory Group theory additive group theory combinatorial number theory factorization number theory sumsets

#### Authors and affiliations

• Alfred Geroldinger
• 1
• Imre Z. Ruzsa
• 2
1. 1.Institute for Mathematics and Scientific ComputingUniversity of GrazGrazAustria
2. 2.Alfréd Rényi Institute of MathematicsBudapestHungary

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-7643-8962-8
• Copyright Information Birkhäuser Basel 2009
• Publisher Name Birkhäuser Basel
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-7643-8961-1
• Online ISBN 978-3-7643-8962-8