The Green–Tao theorem, proved by Ben Green and Terence Tao in 2004,[3] states that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
In 2006, Terence Tao and Tamar Ziegler extended the result to cover polynomial progressions.
The special case when the polynomials are m, 2m, ..., km implies the previous result that there are length k arithmetic progressions of primes.
The Breuillard–Green–Tao theorem, proved by Emmanuel Breuillard, Ben Green, and Terence Tao in 2011,[5] gives a complete classification of approximate groups.
The sets being studied may also be subsets of algebraic structures other than the integers, for example, groups, rings and fields.