Additive Combinatorics

Cambridge Studies in Advanced Mathematics #105

by Terence Tao, Van H. Vu

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.

Genres: Mathematics, Number Theory

532 pages, Hardcover

First published January 1, 2006

About the author

Terence "Terry" Tao FAA FRS (simplified Chinese: 陶哲轩; traditional Chinese: 陶哲軒; pinyin: Táo Zhéxuān) is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing and analytic number theory. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics.

Reviews

[Dilbertario · Rating 5 out of 5]

Libro que merece ser leído en sus partes o en su todo hasta donde la comprensión lo permita.

Desde cuestiones introductorias y asimilables para cualquier estudiante universitario hasta los resultados más profundos y "recientes" sobre primalidad y progresiones aritméticas, con toda la maquinaria que hay detrás desde el cálculo, hasta la estadística más allá de la simple teoría de números.

Un libro del "mejor" matemático vivo según muchos por profundidad y amplitud en el que su estilo superdidáctico queda de manifiesto igual que en su blog personal.

[Pietro · Rating 5 out of 5]

The material is brilliantly motivated, and intuition all but oozes out of its pages. (One has come to expect no less of Tao and Vu.) It's also goddamn *interesting*. The proofs are beautiful, and the exercises incredible. Some challenge you to derive results from research papers!

One thing though; there are a *lot* of typos. Like one every five pages. Some fairly serious, like in equations. Fortunately, Terry Tao has an errata sheet on his webpage.