In additive combinatorics, Freiman's theorem is a central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if is small, then can be contained in a small generalized arithmetic progression. Statement If is a finite subset of with , then is contained in a generalized arithmetic progression of dimension at most and size at most , where and are constants depending only on . Examples For a finite set of integers, it is always true that with equality precisely when is an arithmetic progression. More generally, suppose is a subset of a finite proper generalized arithmetic progression of dimension such that for some real . Then , so that History of Freiman's theorem 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. The current best bounds were provided by Tom Sanders. Tools used in the proof The proof presented here follows the proof in Yufei Zhao's lecture notes. Plünnecke-Ruzsa inequality Ruzsa covering lemma The Ruzsa covering lemma states the following: Let and be finite subsets of an abelian group with nonempty, and let be a positive real number. Then if , there is a subset of with at most elements such that . This lemma provides a bound on how many copies of one needs to cover , hence the name. The proof is essentially a greedy algorithm: Proof: Let be a maximal subset of such that the sets for are all disjoint. Then , and also , so . Furthermore, for any , there is some such that intersects , as otherwise adding to contradicts the maximality of . Thus , so . Freiman homomorphisms and the Ruzsa modeling lemma Let be a positive integer, and and be abelian groups. Let and . A map is a Freiman -homomorphism if whenever for any . If in addition is a bijection and is a Freiman -homomorphism, then is a Freiman -isomorphism. If is a Freiman -homomorphism, then is a Freiman -homomorphism for any positive integer such that . Then the Ruzsa modeling lemma states the following: Let be a finite set of integers, and let be a positive integer. Let be a positive integer such that . Then there