In additive combinatorics, a discipline within mathematics, Freiman's theorem is a central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if |A+A|/|A| is small, then A can be contained in a small generalized arithmetic progression.

Statement

If A is a finite subset of \mathbb{Z} with, then A is contained in a generalized arithmetic progression of dimension at most d(K) and size at most f(K)|A|, where d(K) and f(K) are constants depending only on K.

Examples

For a finite set A of integers, it is always true that with equality precisely when A is an arithmetic progression. More generally, suppose A is a subset of a finite proper generalized arithmetic progression P of dimension d such that for some real C \ge 1. 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 (1992,1994). 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: This lemma provides a bound on how many copies of S-S one needs to cover A, hence the name. The proof is essentially a greedy algorithm: Proof: Let T be a maximal subset of A such that the sets t+S for A are all disjoint. Then, and also , so |T| \le K. Furthermore, for any a \in A, there is some t \in T such that t+S intersects a+S, as otherwise adding a to T contradicts the maximality of T. Thus a \in T+S-S, so.

Freiman homomorphisms and the Ruzsa modeling lemma

Let s \ge 2 be a positive integer, and \Gamma and \Gamma' be abelian groups. Let and. A map is a Freiman s-homomorphism if whenever for any. If in addition \varphi is a bijection and is a Freiman s-homomorphism, then \varphi is a Freiman s-isomorphism. If \varphi is a Freiman s-homomorphism, then \varphi is a Freiman t-homomorphism for any positive integer t such that. Then the Ruzsa modeling lemma states the following: The last statement means there exists some Freiman s-homomorphism between the two subsets. Proof sketch: Choose a prime q sufficiently large such that the modulo-q reduction map is a Freiman s-isomorphism from A 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 \lambda map, which is a Freiman s-isomorphism. Let B be the image. Choose a suitable subset B' of B with cardinality at least |B|/s such that the restriction of \psi_q to B' is a Freiman s-isomorphism onto its image, and let be the preimage of B' under. Then the restriction of to A' is a Freiman s-isomorphism onto its image \psi_q(B'). Lastly, there exists some choice of nonzero \lambda such that the restriction of the modulo-N reduction to \psi_q(B') is a Freiman s-isomorphism onto its image. The result follows after composing this map with the earlier Freiman s-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: Generalizing this lemma to arbitrary cyclic groups requires an analogous notion to “subspace”: that of the Bohr set. Let R be a subset of where N is a prime. The Bohr set of dimension |R| and width \varepsilon is where |rx/N| is the distance from rx/N to the nearest integer. The following lemma generalizes Bogolyubov's lemma: 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. 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 a prime N such that. By the Ruzsa modeling lemma, there exists a subset A' of A of cardinality at least |A|/8 such that A' is Freiman 8-isomorphic to a subset. By the generalization of Bogolyubov's lemma, 2B-2B contains a proper generalized arithmetic progression of dimension d at most and size at least (1/(4d))^dN. Because A' and B are Freiman 8-isomorphic, 2A'-2A' and 2B-2B are Freiman 2-isomorphic. Then the image under the 2-isomorphism of the proper generalized arithmetic progression in 2B-2B is a proper generalized arithmetic progression in called P. But, since. Thus so by the Ruzsa covering lemma for some of cardinality at most (4d)^d. Then X+P-P is contained in a generalized arithmetic progression of dimension |X|+d 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 G is a set P+H for a proper generalized arithmetic progression P and a subgroup H of G. The dimension of this coset progression is defined to be the dimension of P, and its size is defined to be the cardinality of the whole set. Green and Ruzsa showed the following: Green and Ruzsa provided upper bounds of and for some absolute constant C. 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 K<2, when a set has very small doubling, are referred to as Kneser theorems. The polynomial Freiman–Ruzsa conjecture, is a generalization published in a paper by Imre Ruzsa but credited by him to Katalin Marton. It states that if a subset of a group (a power of a cyclic group) A\subset G has doubling constant such that then it is covered by a bounded number K^{C}of cosets of some subgroup H\subset G with|H|\le |A|. In 2012 Tom Sanders gave an almost polynomial bound of the conjecture for abelian groups. In 2023 a solution over a field of characteristic 2 has been posted as a preprint by Tim Gowers, Ben Green, Freddie Manners and Terry Tao. This proof was completely formalized in the Lean 4 formal proof language, a collaborative project that marked an important milestone in terms of mathematicians successfully formalizing contemporary mathematics.