# Freiman's theorem

In mathematics, Freiman's theorem is a combinatorial result in additive number theory. In a sense it accounts for the approximate structure of sets of integers that contain a high proportion of their internal sums, taken two at a time.

The formal statement is:

Let A be a finite set of integers such that the sumset

$A+A$ is small, in the sense that

$|A+A| for some constant $c$ . There exists an n-dimensional arithmetic progression of length

$c'|A|$ that contains A, and such that c' and n depend only on c.

A simple instructive case is the following. We always have

$|A+A|\geq 2|A|-1$ with equality precisely when A is an arithmetic progression.

This result is due to Gregory Freiman (1964,1966). Much interest in it, and applications, stemmed from a new proof by Imre Z. Ruzsa (1994). Mei-Chu Chang proved new polynomial estimates for the size of arithmetic progressions arising in the theorem in 2002.

Green and Ruzsa (2007) generalized the theorem for arbitrary abelian groups: here A can be contained in the sum of a generalized arithmetic progression and a subgroup — the name of such sets is coset-progression.