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Combinatorial Number Theory and Additive Group Theory

Author: Manuel Castellet; Imre Z Ruzsa; Alfred Geroldinger
Publisher: Basel : Birkhäuser Basel, 2009.
Series: Advanced Courses in Mathematics - CRM Barcelona, Centre de Recerca Matemàtica.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
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  Read more...
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Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Manuel Castellet; Imre Z Ruzsa; Alfred Geroldinger
ISBN: 9783764389628 3764389621
OCLC Number: 428884873
Description: 1 online resource (: v.: digital.
Contents: Foreword --
Preface --
I. Additive Group Theory and Non-unique Factorizations --
II. Sumsets and structure --
III. Thematic seminars --
Contributions by Jean Marc Deshouillers, Christian Elsholtz, Gregory A. Freiman, Yahya O. Hamidoune, Norbert Hegyvári, Gyula Károly, Melvyn B. Nathanson, Jaroslav Nešetril, Jozsef Solymosi, Yonutz V. Stanchescu, Gilles Zémor.
Series Title: Advanced Courses in Mathematics - CRM Barcelona, Centre de Recerca Matemàtica.
Responsibility: by Manuel Castellet, Alfred Geroldinger, Imre Z. Ruzsa.

Abstract:

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{u2019}s graph theoretical method), Freiman{u2019}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.

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