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
Szemeredi'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.
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