Additive combinatorics

Summary

Additive combinatorics studies finite subsets of abelian groups, such as sets of integers, and how they behave under addition. Typical questions ask how large a sumset $A + B$ must be, and what a set with a small sumset looks like. Others ask when a set contains additive patterns such as arithmetic progressions. It combines combinatorics, Fourier analysis, ergodic theory and number theory. Its results include Szemerédi's theorem, Freiman's theorem and the Green–Tao theorem.