Developments in Mathematics

Structural Additive Theory

Authors: Grynkiewicz, David

  • ​Focuses on areas of Additive Theory that have not been treated in detail in book form
  • Requires little pre-requisite apart from a solid background in undergraduate mathematics and graduate-level introductory algebra
  • Contains detail-rich proofs, making the material more accessible to newcomers in the field
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eBook $109.00
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  • ISBN 978-3-319-00416-7
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  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-3-319-00415-0
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  • Usually dispatched within 3 to 5 business days.
Softcover $149.00
price for USA
  • ISBN 978-3-319-37518-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
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About this book

​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. 

Reviews

From the book reviews:

“This volume is a very welcome addition to the rather small body of books published in the area of additive combinatorics so far. … Presenting a systematic treatment of a surprisingly wide range of topics from additive combinatorics, this book is perfectly suited as a textbook for a course in additive combinatorics … . In many cases, the material included has not appeared in book form before and can only be found in research papers.” (Vsevolod F. Lev, Mathematical Reviews, April, 2014)

“The book contains several recently developed theories that the author puts into Additive Theory … . The author has added some scrumptious exercises at the end of every chapter, and supplementary notes for interested readers. … can be used as a text at the graduate level, mainly to introduce essential topics of the ‘Additive Theory’, and to show to students and anybody interested to do to the research in this field, what is going on and what they can do.” (Mehdi Hassani, MAA Reviews, February, 2014)


Table of contents (22 chapters)

  • Abelian Groups and Character Sums

    Grynkiewicz, David J.

    Pages 1-10

  • Introduction to Sumsets

    Grynkiewicz, David J.

    Pages 13-23

  • Simple Results for Torsion-Free Abelian Groups

    Grynkiewicz, David J.

    Pages 25-28

  • Basic Results for Sumsets with an Infinite Summand

    Grynkiewicz, David J.

    Pages 29-56

  • The Pigeonhole and Multiplicity Bounds

    Grynkiewicz, David J.

    Pages 57-60

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-3-319-00416-7
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-3-319-00415-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $149.00
price for USA
  • ISBN 978-3-319-37518-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Structural Additive Theory
Authors
Series Title
Developments in Mathematics
Series Volume
30
Copyright
2013
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-00416-7
DOI
10.1007/978-3-319-00416-7
Hardcover ISBN
978-3-319-00415-0
Softcover ISBN
978-3-319-37518-2
Series ISSN
1389-2177
Edition Number
1
Number of Pages
XII, 426
Topics