Structural Additive Theory, 1st Edition

  • Published By:
  • ISBN-10: 3319004166
  • ISBN-13: 9783319004167
  • DDC: 511.6
  • Grade Level Range: College Freshman - College Senior
  • 426 Pages | eBook
  • Original Copyright 2013 | Published/Released June 2014
  • This publication's content originally published in print form: 2013

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​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. 

Table of Contents

Front Cover.
Other Frontmatter.
Title Page.
Copyright Page.
1: Abelian Groups and Character Sums.
2: Sumsets.
3: Introduction to Sumsets.
4: Simple Results for Torsion-Free Abelian Groups.
5: Basic Results for Sumsets with an Infinite Summand.
6: The Pigeonhole and Multiplicity Bounds.
7: Periodic Sets and Kneser’s Theorem.
8: Compression, Complements and the 3k − 4 Theorem.
9: Additive Energy.
10: Kemperman’s Critical Pair Theory.
11: Subsequence Sums.
12: Zero-Sums, Setpartitions and Subsequence Sums.
13: Long Zero-Sum Free Sequences over Cyclic Groups.
14: Pollard’s Theorem for General Abelian Groups.
15: The DeVos-Goddyn-Mohar Theorem.
16: The Partition Theorem I.
17: The Partition Theorem II.
18: The ψWeighted Gao Theorem.
19: Advanced Methods.
20: Group Algebras: An Upper Bound for the Davenport Constant.
21: Character and Linear Algebraic Methods: Snevily’s Conjecture.
22: Character Sum and Fourier Analytic Methods: r-Critical Pairs I.
23: Freiman Homomorphisms Revisited.
24: The Isoperimetric Method: Sidon Sets and r-Critical Pairs II.
25: The Polynomial Method: The Erdős-Heilbronn Conjecture.