Structural Additive Theory

  • David J. Grynkiewicz

Part of the Developments in Mathematics book series (DEVM, volume 30)

Table of contents

  1. Front Matter
    Pages I-XII
  2. David J. Grynkiewicz
    Pages 1-10
  3. Sumsets

    1. Front Matter
      Pages 11-11
    2. David J. Grynkiewicz
      Pages 13-23
    3. David J. Grynkiewicz
      Pages 25-28
    4. David J. Grynkiewicz
      Pages 29-56
    5. David J. Grynkiewicz
      Pages 57-60
    6. David J. Grynkiewicz
      Pages 61-69
    7. David J. Grynkiewicz
      Pages 71-98
    8. David J. Grynkiewicz
      Pages 99-109
    9. David J. Grynkiewicz
      Pages 111-132
  4. Subsequence Sums

    1. Front Matter
      Pages 133-133
    2. David J. Grynkiewicz
      Pages 135-144
    3. David J. Grynkiewicz
      Pages 145-153
    4. David J. Grynkiewicz
      Pages 155-179
    5. David J. Grynkiewicz
      Pages 181-195
    6. David J. Grynkiewicz
      Pages 197-227
    7. David J. Grynkiewicz
      Pages 229-244
    8. David J. Grynkiewicz
      Pages 245-262
  5. Advanced Methods

    1. Front Matter
      Pages 263-263

About this book

Introduction

Nestled between number theory, combinatorics, algebra, and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e. sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this book usable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field.


The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune’s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions.

 

Keywords

Abelian groups Freiman homomorphisms Kneser's Theorem Universal Ambient Group of sumsets Vosper's Theorem additive combinatorics additive number theory subsequence sums

Authors and affiliations

  • David J. Grynkiewicz
    • 1
  1. 1., Institut für MathematikKarl-Franzens-Universität GrazGrazAustria

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-00416-7
  • Copyright Information Springer International Publishing Switzerland 2013
  • Publisher Name Springer, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-00415-0
  • Online ISBN 978-3-319-00416-7
  • Series Print ISSN 1389-2177
  • About this book
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