Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory.
This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects thenotes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.

Additive Group Theory and Non-unique Factorizations.- Notation.- Basic concepts of non-unique factorizations.- The Davenport constant and first precise arithmetical results.- The structure of sets of lengths.- Addition theorems and direct zero-sum problems.- Inverse zero-sum problems and arithmetical consequences.- Sumsets and Structure.- Notation.- Cardinality inequalities.- Structure of sets with few sums.- Location and sumsets.- Density.- Measure and topology.- Exercises.- Thematic seminars.- A survey on additive and multiplicative decompositions of sumsets and of shifted sets.- On the detailed structure of sets with small additive property.- The isoperimetric method.- Additive structure of difference sets.- The polynomial method in additive combinatorics.- Problems in additive number theory, III.- Incidences and the spectra of graphs.- Multi-dimensional inverse additive problems.