a prime such that . By the Ruzsa modeling lemma, there exists a subset of of cardinality at least such that is Freiman 8-isomorphic to a subset . By the generalization of Bogolyubov's lemma, contains a proper generalized arithmetic progression of dimension at most and size at least . Because and are Freiman 8-isomorphic, and are Freiman 2-isomorphic. Then the image under the 2-isomorphism of the proper generalized arithmetic progression in is a proper generalized arithmetic progression in called . But , since . Thus so by the Ruzsa covering lemma for some of cardinality at most . Then is contained in a generalized arithmetic progression of dimension and size at most , completing the proof. Generalizations A result due to Ben Green and Imre Ruzsa generalized Freiman's theorem to arbitrary abelian groups. They used an analogous notion to generalized arithmetic progressions, which they called coset progressions. A coset progression''' of an abelian group is a set for a proper generalized arithmetic progression and a subgroup of . The dimension of this coset progression is defined to be the dimension of , and its size is defined to be the cardinality of the whole set. Green and Ruzsa showed the following: Let be a finite set in an abelian group such that . Then is contained in a coset progression of dimension at most and size at most , where and are functions of that are independent of . Green and Ruzsa provided upper bounds of and for some absolute constant . Terence Tao (2010) also generalized Freiman's theorem to solvable groups of bounded derived length. Extending Freiman’s theorem to an arbitrary nonabelian group is still open. Results for , when a set has very small doubling, are referred to as Kneser theorems. See also Markov spectrum Plünnecke-Ruzsa inequality Kneser's theorem (combinatorics) References Further reading Sumsets Theorems in number theory