Additive Combinatorics

A graduate-level 2006 text bringing together the tools from different fields used in additive combinatorics.

Terence Tao (Author), Van H. Vu (Author)

9780521136563, Cambridge University Press

Paperback, published 19 November 2009

532 pages, 640 exercises
22.9 x 15.2 x 3 cm, 0.75 kg

'The book gathers diverse important techniques used in additive combinatorics, and its main advantage is that it is written in a very readable and easy to understand style. The authors try very successfully to develop all the necessary background material … [which] makes the book useful not only to graduate students, but also to researchers who are interested to learn more about the variety of diverse tools and ideas applied in this fascinating subject.' Zentralblatt MATH

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 2006 text will allow students and researchers easy entry into this fascinating field. Here, 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 Szemerédi'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 sum set theorems
6. Graph-theoretic methods
7. The Littlewood–Offord problem
8. Incidence geometry
9. Algebraic methods
10. Szemerédi's theorem for k = 3
11. Szemerédi's theorem for k > 3
12. Long arithmetic progressions in sum sets
Bibliography
Index.

Subject Areas: Combinatorics & graph theory [PBV]