A graduate level textbook bringing together the tools from different fields used in additive combinatorics.

Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate level textbook will allow students and researchers easy entry into this fascinating field. Here, for the first time, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemer 's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.

Prologue
1. The probabilistic method
2. Sum set estimates
3. Additive geometry
4. Fourier analytic methods
5. Inverse sumset theorems
6. Graph theoretic methods
7. The Littlewood-Offord problem
8. Incidence geometry
9. Algebraic methods
10. Szemer 's theorem for k = 3
11. Szemer 's theorem for k >
3
12. Long arithmetic progressions in sumsets
Bibliography.