# 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

${\displaystyle A+A}$

is small, in the sense that

${\displaystyle |A+A|

for some constant ${\displaystyle c}$. There exists an n-dimensional arithmetic progression of length

${\displaystyle c'|A|}$

that contains A, and such that c' and n depend only on c.[1]

A simple instructive case is the following. We always have

${\displaystyle |A+A|\geq 2|A|-1}$

with equality precisely when A is an arithmetic progression.

This result is due to Gregory Freiman (1964,1966).[2] 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.[3]

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.