# Freiman's theorem

Freiman's theorem 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 {displaystyle |A+A|/|A|} is small, then {displaystyle A} can be contained in a small generalized arithmetic progression.

Contents 1 Statement 2 Examples 3 History of Freiman's theorem 4 Tools used in the proof 4.1 Plünnecke-Ruzsa inequality 4.2 Ruzsa covering lemma 4.3 Freiman homomorphisms and the Ruzsa modeling lemma 4.4 Bohr sets and Bogolyubov's lemma 5 Proof 6 Generalizations 7 See also 8 References 9 Further reading Statement If {displaystyle A} is a finite subset of {displaystyle mathbb {Z} } with {displaystyle |A+A|leq K|A|} , then {displaystyle A} is contained in a generalized arithmetic progression of dimension at most {displaystyle d(K)} and size at most {displaystyle f(K)|A|} , where {displaystyle d(K)} and {displaystyle f(K)} are constants depending only on {displaystyle K} .

Examples For a finite set {displaystyle A} of integers, it is always true that {displaystyle |A+A|geq 2|A|-1,} with equality precisely when {displaystyle A} is an arithmetic progression.

More generally, suppose {displaystyle A} is a subset of a finite proper generalized arithmetic progression {displaystyle P} of dimension {displaystyle d} such that {displaystyle |P|leq C|A|} for some real {displaystyle Cgeq 1} . Then {displaystyle |P+P|leq 2^{d}|P|} , so that {displaystyle |A+A|leq |P+P|leq 2^{d}|P|leq C2^{d}|A|.} History of Freiman's theorem This result is due to Gregory Freiman (1964, 1966).[1][2][3] Much interest in it, and applications, stemmed from a new proof by Imre Z. Ruzsa (1994).[4] Mei-Chu Chang proved new polynomial estimates for the size of arithmetic progressions arising in the theorem in 2002.[5] The current best bounds were provided by Tom Sanders.[6] Tools used in the proof The proof presented here follows the proof in Yufei Zhao's lecture notes.[7] Plünnecke-Ruzsa inequality Main article: Plünnecke-Ruzsa inequality Ruzsa covering lemma The Ruzsa covering lemma states the following: Let {displaystyle A} and {displaystyle S} be finite subsets of an abelian group with {displaystyle S} nonempty, and let {displaystyle K} be a positive real number. Then if {displaystyle |A+S|leq K|S|} , there is a subset {displaystyle T} of {displaystyle A} with at most {displaystyle K} elements such that {displaystyle Asubseteq T+S-S} .

This lemma provides a bound on how many copies of {displaystyle S-S} one needs to cover {displaystyle A} , hence the name. The proof is essentially a greedy algorithm: Proof: Let {displaystyle T} be a maximal subset of {displaystyle A} such that the sets {displaystyle t+S} for {displaystyle A} are all disjoint. Then {displaystyle |T+S|=|T|cdot |S|} , and also {displaystyle |T+S|leq |A+S|leq K|S|} , so {displaystyle |T|leq K} . Furthermore, for any {displaystyle ain A} , there is some {displaystyle tin T} such that {displaystyle t+S} intersects {displaystyle a+S} , as otherwise adding {displaystyle a} to {displaystyle T} contradicts the maximality of {displaystyle T} . Thus {displaystyle ain T+S-S} , so {displaystyle Asubseteq T+S-S} .

Freiman homomorphisms and the Ruzsa modeling lemma Let {displaystyle sgeq 2} be a positive integer, and {displaystyle Gamma } and {displaystyle Gamma '} be abelian groups. Let {displaystyle Asubseteq Gamma } and {displaystyle Bsubseteq Gamma '} . A map {displaystyle varphi colon Ato B} is a Freiman {displaystyle s} -homomorphism if {displaystyle varphi (a_{1})+cdots +varphi (a_{s})=varphi (a_{1}')+cdots +varphi (a_{s}')} whenever {displaystyle a_{1}+cdots +a_{s}=a_{1}'+cdots +a_{s}'} for any {displaystyle a_{1},ldots ,a_{s},a_{1}',ldots ,a_{s}'in A} .

If in addition {displaystyle varphi } is a bijection and {displaystyle varphi ^{-1}colon Bto A} is a Freiman {displaystyle s} -homomorphism, then {displaystyle varphi } is a Freiman {displaystyle s} -isomorphism.

If {displaystyle varphi } is a Freiman {displaystyle s} -homomorphism, then {displaystyle varphi } is a Freiman {displaystyle t} -homomorphism for any positive integer {displaystyle t} such that {displaystyle 2leq tleq s} .

Then the Ruzsa modeling lemma states the following: Let {displaystyle A} be a finite set of integers, and let {displaystyle sgeq 2} be a positive integer. Let {displaystyle N} be a positive integer such that {displaystyle Ngeq |sA-sA|} . Then there exists a subset {displaystyle A'} of {displaystyle A} with cardinality at least {displaystyle |A|/s} such that {displaystyle A'} is Freiman {displaystyle s} -isomorphic to a subset of {displaystyle mathbb {Z} /Nmathbb {Z} } .

The last statement means there exists some Freiman {displaystyle s} -homomorphism between the two subsets.

Proof sketch: Choose a prime {displaystyle q} sufficiently large such that the modulo- {displaystyle q} reduction map {displaystyle pi _{q}colon mathbb {Z} to mathbb {Z} /qmathbb {Z} } is a Freiman {displaystyle s} -isomorphism from {displaystyle A} to its image in {displaystyle mathbb {Z} /qmathbb {Z} } . Let {displaystyle psi _{q}colon mathbb {Z} /qmathbb {Z} to mathbb {Z} } be the lifting map that takes each member of {displaystyle mathbb {Z} /qmathbb {Z} } to its unique representative in {displaystyle {1,ldots ,q}subseteq mathbb {Z} } . For nonzero {displaystyle lambda in mathbb {Z} /qmathbb {Z} } , let {displaystyle cdot lambda colon mathbb {Z} /qmathbb {Z} to mathbb {Z} /qmathbb {Z} } be the multiplication by {displaystyle lambda } map, which is a Freiman {displaystyle s} -isomorphism. Let {displaystyle B} be the image {displaystyle (cdot lambda circ pi _{q})(A)} . Choose a suitable subset {displaystyle B'} of {displaystyle B} with cardinality at least {displaystyle |B|/s} such that the restriction of {displaystyle psi _{q}} to {displaystyle B'} is a Freiman {displaystyle s} -isomorphism onto its image, and let {displaystyle A'subseteq A} be the preimage of {displaystyle B'} under {displaystyle cdot lambda circ pi _{q}} . Then the restriction of {displaystyle psi _{q}circ cdot lambda circ pi _{q}} to {displaystyle A'} is a Freiman {displaystyle s} -isomorphism onto its image {displaystyle psi _{q}(B')} . Lastly, there exists some choice of nonzero {displaystyle lambda } such that the restriction of the modulo- {displaystyle N} reduction {displaystyle mathbb {Z} to mathbb {Z} /Nmathbb {Z} } to {displaystyle psi _{q}(B')} is a Freiman {displaystyle s} -isomorphism onto its image. The result follows after composing this map with the earlier Freiman {displaystyle 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: Let {displaystyle Ain mathbb {F} _{2}^{n}} and let {displaystyle alpha =|A|/2^{n}} . Then {displaystyle 4A} contains a subspace of {displaystyle mathbb {F} _{2}^{n}} of dimension at least {displaystyle n-alpha ^{-2}} .

Generalizing this lemma to arbitrary cyclic groups requires an analogous notion to “subspace”: that of the Bohr set. Let {displaystyle R} be a subset of {displaystyle mathbb {Z} /Nmathbb {Z} } where {displaystyle N} is a prime. The Bohr set of dimension {displaystyle |R|} and width {displaystyle epsilon } is {displaystyle operatorname {Bohr} (R,epsilon )={xin mathbb {Z} /Nmathbb {Z} :forall rin R,|rx/N|leq epsilon },} where {displaystyle |rx/N|} is the distance from {displaystyle rx/N} to the nearest integer. The following lemma generalizes Bogolyubov's lemma: Let {displaystyle Ain mathbb {Z} /Nmathbb {Z} } and {displaystyle alpha =|A|/N} . Then {displaystyle 2A-2A} contains a Bohr set of dimension at most {displaystyle alpha ^{-2}} and width {displaystyle 1/4} .

Here the dimension of a Bohr set is analogous to the codimension of a set in {displaystyle mathbb {F} _{2}^{n}} . 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 {displaystyle X} be a Bohr set in {displaystyle mathbb {Z} /Nmathbb {Z} } of dimension {displaystyle d} and width {displaystyle epsilon } . Then {displaystyle X} contains a proper generalized arithmetic progression of dimension at most {displaystyle d} and size at least {displaystyle (epsilon /d)^{d}N} .

The proof of this proposition uses Minkowski's theorem, a fundamental result in geometry of numbers.

Proof By the Plünnecke-Ruzsa inequality, {displaystyle |8A-8A|leq K^{16}|A|} . By Bertrand's postulate, there exists a prime {displaystyle N} such that {displaystyle |8A-8A|leq Nleq 2K^{16}|A|} . By the Ruzsa modeling lemma, there exists a subset {displaystyle A'} of {displaystyle A} of cardinality at least {displaystyle |A|/8} such that {displaystyle A'} is Freiman 8-isomorphic to a subset {displaystyle Bsubseteq mathbb {Z} /Nmathbb {Z} } .

By the generalization of Bogolyubov's lemma, {displaystyle 2B-2B} contains a proper generalized arithmetic progression of dimension {displaystyle d} at most {displaystyle (1/(8cdot 2K^{16}))^{-2}=256K^{32}} and size at least {displaystyle (1/(4d))^{d}N} . Because {displaystyle A'} and {displaystyle B} are Freiman 8-isomorphic, {displaystyle 2A'-2A'} and {displaystyle 2B-2B} are Freiman 2-isomorphic. Then the image under the 2-isomorphism of the proper generalized arithmetic progression in {displaystyle 2B-2B} is a proper generalized arithmetic progression in {displaystyle 2A'-2A'subseteq 2A-2A} called {displaystyle P} .

But {displaystyle P+Asubseteq 3A-2A} , since {displaystyle Psubseteq 2A-2A} . Thus {displaystyle |P+A|leq |3A-2A|leq |8A-8A|leq Nleq (4d)^{d}|P|} so by the Ruzsa covering lemma {displaystyle Asubseteq X+P-P} for some {displaystyle Xsubseteq A} of cardinality at most {displaystyle (4d)^{d}} . Then {displaystyle X+P-P} is contained in a generalized arithmetic progression of dimension {displaystyle |X|+d} and size at most {displaystyle 2^{|X|}2^{d}|P|leq 2^{|X|+d}|2A-2A|leq 2^{|X|+d}K^{4}|A|} , 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 {displaystyle G} is a set {displaystyle P+H} for a proper generalized arithmetic progression {displaystyle P} and a subgroup {displaystyle H} of {displaystyle G} . The dimension of this coset progression is defined to be the dimension of {displaystyle P} , and its size is defined to be the cardinality of the whole set. Green and Ruzsa showed the following: Let {displaystyle A} be a finite set in an abelian group {displaystyle G} such that {displaystyle |A+A|leq K|A|} . Then {displaystyle A} is contained in a coset progression of dimension at most {displaystyle d(K)} and size at most {displaystyle f(K)|A|} , where {displaystyle f(K)} and {displaystyle d(K)} are functions of {displaystyle K} that are independent of {displaystyle G} .