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. Bohr sets and Bogolyubov's lemma 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 . The proof of this proposition uses Minkowski's theorem, a fundamental result in geometry of numbers. Proof By the Plünnecke-Ruzsa inequality, . By Bertrand's postulate, there exists