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.
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