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
is small, in the sense that
for some constant . There exists an n-dimensional arithmetic progression of length
that contains A, and such that c' and n depend only on c.
A simple instructive case is the following. We always have
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.
- Freiman, G.A. (1964). "Addition of finite sets". Sov. Math., Dokl. 5: 1366–1370. Zbl 0163.29501.
- Freiman, G. A. (1966). Foundations of a Structural Theory of Set Addition (in Russian). Kazan: Kazan Gos. Ped. Inst. p. 140. Zbl 0203.35305.
- Freiman, G. A. (1999). "Structure theory of set addition". Astérisque. 258: 1–33. Zbl 0958.11008.
- Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and Geometry of Sumsets. Graduate Texts in Mathematics. 165. Springer. ISBN 0-387-94655-1. Zbl 0859.11003.
- Ruzsa, Imre Z. (1994). "Generalized arithmetical progressions and sumsets". Acta Mathematica Hungarica. 65 (4): 379–388. doi:10.1007/bf01876039. Zbl 0816.11008.
- Ruzsa, Imre Z.; Green, Ben (2007). "Freiman's theorem in an arbitrary abelian group". London Math. Soc. 75 (1): 163–175. doi:10.1112/jlms/jdl021.