Ebook: Structural Additive Theory

​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.

---

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.

---

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.

Content:
Front Matter....Pages I-XII
Front Matter....Pages 11-11
Introduction to Sumsets....Pages 13-23
Simple Results for Torsion-Free Abelian Groups....Pages 25-28
Basic Results for Sumsets with an Infinite Summand....Pages 29-56
The Pigeonhole and Multiplicity Bounds....Pages 57-60
Periodic Sets and Kneser’s Theorem....Pages 61-69
Compression, Complements and the 3k?4 Theorem....Pages 71-98
Additive Energy....Pages 99-109
Kemperman’s Critical Pair Theory....Pages 111-132
Front Matter....Pages 133-133
Zero-Sums, Setpartitions and Subsequence Sums....Pages 135-144
Long Zero-Sum Free Sequences over Cyclic Groups....Pages 145-153
Pollard’s Theorem for General Abelian Groups....Pages 155-179
The DeVos-Goddyn-Mohar Theorem....Pages 181-195
The Partition Theorem I....Pages 197-227
The Partition Theorem II....Pages 229-244
The ?-Weighted Gao Theorem....Pages 245-262
Front Matter....Pages 263-263
Group Algebras: An Upper Bound for the Davenport Constant....Pages 265-270
Character and Linear Algebraic Methods: Snevily’s Conjecture....Pages 271-278
Character Sum and Fourier Analytic Methods: r-Critical Pairs I....Pages 279-298
Freiman Homomorphisms Revisited....Pages 299-365
Abelian Groups and Character Sums....Pages 1-10
Front Matter....Pages 263-263
The Isoperimetric Method: Sidon Sets and r-Critical Pairs II....Pages 367-400
The Polynomial Method: The Erd?s-Heilbronn Conjecture....Pages 401-414
Back Matter....Pages 415-426

---

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.

Content:
Front Matter....Pages I-XII
Front Matter....Pages 11-11
Introduction to Sumsets....Pages 13-23
Simple Results for Torsion-Free Abelian Groups....Pages 25-28
Basic Results for Sumsets with an Infinite Summand....Pages 29-56
The Pigeonhole and Multiplicity Bounds....Pages 57-60
Periodic Sets and Kneser’s Theorem....Pages 61-69
Compression, Complements and the 3k?4 Theorem....Pages 71-98
Additive Energy....Pages 99-109
Kemperman’s Critical Pair Theory....Pages 111-132
Front Matter....Pages 133-133
Zero-Sums, Setpartitions and Subsequence Sums....Pages 135-144
Long Zero-Sum Free Sequences over Cyclic Groups....Pages 145-153
Pollard’s Theorem for General Abelian Groups....Pages 155-179
The DeVos-Goddyn-Mohar Theorem....Pages 181-195
The Partition Theorem I....Pages 197-227
The Partition Theorem II....Pages 229-244
The ?-Weighted Gao Theorem....Pages 245-262
Front Matter....Pages 263-263
Group Algebras: An Upper Bound for the Davenport Constant....Pages 265-270
Character and Linear Algebraic Methods: Snevily’s Conjecture....Pages 271-278
Character Sum and Fourier Analytic Methods: r-Critical Pairs I....Pages 279-298
Freiman Homomorphisms Revisited....Pages 299-365
Abelian Groups and Character Sums....Pages 1-10
Front Matter....Pages 263-263
The Isoperimetric Method: Sidon Sets and r-Critical Pairs II....Pages 367-400
The Polynomial Method: The Erd?s-Heilbronn Conjecture....Pages 401-414
Back Matter....Pages 415-426
....