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 exists a subset of with cardinality at least such that is Freiman -isomorphic to a subset of .
The last statement means there exists some Freiman -homomorphism between the two subsets.
Proof sketch: Choose a prime sufficiently large such that the modulo- reduction map is a Freiman -isomorphism from to its image in . Let be the lifting map that takes each member of to its unique representative in . For nonzero , let be the multiplication by map, which is a Freiman -isomorphism. Let be the image . Choose a suitable subset of with cardinality at least such that the restriction of to is a Freiman -isomorphism onto its image, and let be the preimage of under . Then the restriction of to is a Freiman -isomorphism onto its image . Lastly, there exists some choice of nonzero such that the restriction of the modulo- reduction to is a Freiman -isomorphism onto its image. The result follows after composing this map with the earlier Freiman -isomorphism.
Though Freiman's theorem applies to sets of integers, the Ruzsa modeling lemma allows one to model sets of integers as subsets of finite cyclic groups. So it is useful to first work in the setting of a finite field, and then generalize results to the integers. The following lemma was proved by Bogolyubov:
Let and let . Then contains a subspace of of dimension at least .
Generalizing this lemma to arbitrary cyclic groups requires an analogous notion to “subspace”: that of the Bohr set. Let be a subset of where is a prime. The Bohr set of dimension and width is
where is the distance from to the nearest integer. The following lemma generalizes Bogolyubov's lemma:
Let and . Then contains a Bohr set of dimension at most and width .
Here the dimension of a Bohr set is analogous to the codimension of a set in . The proof of the lemma involves Fourier-analytic methods. The following proposition relates Bohr sets back to generalized arithmetic progressions, eventually leading to the proof of Freiman's theorem.
Let be a Bohr set in of dimension and width . Then contains a proper generalized arithmetic progression of dimension at most and size at least .
By the Plünnecke-Ruzsa inequality, . By Bertrand's postulate, there exists 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.
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 .