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 {style d'affichage |A+A|/|UN|} is small, alors {style d'affichage A} can be contained in a small generalized arithmetic progression.

Contenu 1 Déclaration 2 Exemples 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 Preuve 6 Généralisations 7 Voir également 8 Références 9 Further reading Statement If {style d'affichage A} is a finite subset of {style d'affichage mathbb {Z} } avec {style d'affichage |A+A|leq K|UN|} , alors {style d'affichage A} is contained in a generalized arithmetic progression of dimension at most {displaystyle d(K)} and size at most {style d'affichage f(K)|UN|} , où {displaystyle d(K)} et {style d'affichage f(K)} are constants depending only on {style d'affichage K} .

Examples For a finite set {style d'affichage A} of integers, it is always true that {style d'affichage |A+A|gq 2|UN|-1,} with equality precisely when {style d'affichage A} is an arithmetic progression.

Plus généralement, supposer {style d'affichage A} is a subset of a finite proper generalized arithmetic progression {style d'affichage P} of dimension {displaystyle d} tel que {style d'affichage |P|leq C|UN|} for some real {displaystyle Cgeq 1} . Alors {style d'affichage |P+P|leq 2^{ré}|P|} , pour que {style d'affichage |A+A|leq |P+P|leq 2^{ré}|P|leq C2^{ré}|UN|.} History of Freiman's theorem This result is due to Gregory Freiman (1964, 1966).[1][2][3] Much interest in it, et 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: Laisser {style d'affichage A} et {style d'affichage S} be finite subsets of an abelian group with {style d'affichage S} nonempty, et laissez {style d'affichage K} be a positive real number. Puis si {style d'affichage |A+S|leq K|S|} , there is a subset {style d'affichage T} de {style d'affichage A} with at most {style d'affichage 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 {style d'affichage A} , hence the name. The proof is essentially a greedy algorithm: Preuve: Laisser {style d'affichage T} be a maximal subset of {style d'affichage A} such that the sets {displaystyle t+S} pour {style d'affichage A} are all disjoint. Alors {style d'affichage |T+S|=|J|cdot |S|} , and also {style d'affichage |T+S|leq |A+S|leq K|S|} , alors {style d'affichage |J|leq K} . Par ailleurs, pour toute {style d'affichage ain A} , there is some {displaystyle tin T} tel que {displaystyle t+S} intersects {displaystyle a+S} , as otherwise adding {style d'affichage a} à {style d'affichage T} contradicts the maximality of {style d'affichage T} . Ainsi {displaystyle ain T+S-S} , alors {displaystyle Asubseteq T+S-S} .

Freiman homomorphisms and the Ruzsa modeling lemma Let {displaystyle sgeq 2} be a positive integer, et {style d'affichage Gamma } et {displaystyle Gamma '} be abelian groups. Laisser {displaystyle Asubseteq Gamma } et {displaystyle Bsubseteq Gamma '} . A map {displaystyle varphi colon Ato B} is a Freiman {style d'affichage s} -homomorphism if {style d'affichage varphi (un_{1})+cdots +varphi (un_{s})=varphi (un_{1}')+cdots +varphi (un_{s}')} chaque fois que {style d'affichage a_{1}+cdots +a_{s}=a_{1}'+cdots +a_{s}'} pour toute {style d'affichage a_{1},ldots ,un_{s},un_{1}',ldots ,un_{s}'in A} .

If in addition {style d'affichage varphi } is a bijection and {displaystyle varphi ^{-1}colon Bto A} is a Freiman {style d'affichage s} -homomorphism, alors {style d'affichage varphi } is a Freiman {style d'affichage s} -isomorphism.

Si {style d'affichage varphi } is a Freiman {style d'affichage s} -homomorphism, alors {style d'affichage varphi } is a Freiman {style d'affichage t} -homomorphism for any positive integer {style d'affichage t} tel que {displaystyle 2leq tleq s} .

Then the Ruzsa modeling lemma states the following: Laisser {style d'affichage A} be a finite set of integers, et laissez {displaystyle sgeq 2} be a positive integer. Laisser {displaystyle N} be a positive integer such that {displaystyle Ngeq |sA-sA|} . Then there exists a subset {style d'affichage A'} de {style d'affichage A} with cardinality at least {style d'affichage |UN|/s} tel que {style d'affichage A'} is Freiman {style d'affichage s} -isomorphic to a subset of {style d'affichage mathbb {Z} /Nmathbb {Z} } .

The last statement means there exists some Freiman {style d'affichage s} -homomorphism between the two subsets.

Esquisse d'épreuve: Choose a prime {style d'affichage q} sufficiently large such that the modulo- {style d'affichage q} reduction map {style d'affichage pi _{q}colon mathbb {Z} à mathbb {Z} /qmathbb {Z} } is a Freiman {style d'affichage s} -isomorphism from {style d'affichage A} to its image in {style d'affichage mathbb {Z} /qmathbb {Z} } . Laisser {style d'affichage psi _{q}colon mathbb {Z} /qmathbb {Z} à mathbb {Z} } be the lifting map that takes each member of {style d'affichage mathbb {Z} /qmathbb {Z} } to its unique representative in {style d'affichage {1,ldots ,q}subseteq mathbb {Z} } . For nonzero {style d'affichage lambda dans mathbb {Z} /qmathbb {Z} } , laisser {displaystyle cdot lambda colon mathbb {Z} /qmathbb {Z} à mathbb {Z} /qmathbb {Z} } be the multiplication by {style d'affichage lambda } map, which is a Freiman {style d'affichage s} -isomorphism. Laisser {style d'affichage B} be the image {style d'affichage (cdot lambda circ pi _{q})(UN)} . Choose a suitable subset {style d'affichage B'} de {style d'affichage B} with cardinality at least {style d'affichage |B|/s} such that the restriction of {style d'affichage psi _{q}} à {style d'affichage B'} is a Freiman {style d'affichage s} -isomorphism onto its image, et laissez {displaystyle A'subseteq A} be the preimage of {style d'affichage B'} under {displaystyle cdot lambda circ pi _{q}} . Then the restriction of {style d'affichage psi _{q}circ cdot lambda circ pi _{q}} à {style d'affichage A'} is a Freiman {style d'affichage s} -isomorphism onto its image {style d'affichage psi _{q}(B')} . Lastly, there exists some choice of nonzero {style d'affichage lambda } such that the restriction of the modulo- {displaystyle N} reduction {style d'affichage mathbb {Z} à mathbb {Z} /Nmathbb {Z} } à {style d'affichage psi _{q}(B')} is a Freiman {style d'affichage s} -isomorphism onto its image. The result follows after composing this map with the earlier Freiman {style d'affichage 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: Laisser {displaystyle Ain mathbb {F} _{2}^{n}} et laissez {displaystyle alpha =|UN|/2^{n}} . Alors {displaystyle 4A} contains a subspace of {style d'affichage 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. Laisser {style d'affichage R} be a subset of {style d'affichage mathbb {Z} /Nmathbb {Z} } où {displaystyle N} is a prime. The Bohr set of dimension {style d'affichage |R|} and width {style d'affichage epsilon } est {nom de l'opérateur de style d'affichage {Bohr} (R,epsilon )={xin mathbb {Z} /Nmathbb {Z} :forall rin R,|rx/N|leq epsilon },} où {style d'affichage |rx/N|} is the distance from {displaystyle rx/N} to the nearest integer. The following lemma generalizes Bogolyubov's lemma: Laisser {displaystyle Ain mathbb {Z} /Nmathbb {Z} } et {displaystyle alpha =|UN|/N} . Alors {displaystyle 2A-2A} contains a Bohr set of dimension at most {displaystyle alpha ^{-2}} and width {style d'affichage 1/4} .

Here the dimension of a Bohr set is analogous to the codimension of a set in {style d'affichage 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.

Laisser {style d'affichage X} be a Bohr set in {style d'affichage mathbb {Z} /Nmathbb {Z} } of dimension {displaystyle d} and width {style d'affichage epsilon } . Alors {style d'affichage X} contains a proper generalized arithmetic progression of dimension at most {displaystyle d} and size at least {style d'affichage (epsilon /d)^{ré}N} .

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

Proof By the Plünnecke-Ruzsa inequality, {style d'affichage |8A-8A|leq K^{16}|UN|} . By Bertrand's postulate, there exists a prime {displaystyle N} tel que {style d'affichage |8A-8A|leq Nleq 2K^{16}|UN|} . By the Ruzsa modeling lemma, there exists a subset {style d'affichage A'} de {style d'affichage A} of cardinality at least {style d'affichage |UN|/8} tel que {style d'affichage 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 {style d'affichage (1/(8cdot 2K^{16}))^{-2}=256K^{32}} and size at least {style d'affichage (1/(4ré))^{ré}N} . Car {style d'affichage A'} et {style d'affichage B} are Freiman 8-isomorphic, {displaystyle 2A'-2A'} et {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} appelé {style d'affichage P} .

Mais {displaystyle P+Asubseteq 3A-2A} , puisque {displaystyle Psubseteq 2A-2A} . Ainsi {style d'affichage |P+A|leq |3A-2A|leq |8A-8A|leq Nleq (4ré)^{ré}|P|} so by the Ruzsa covering lemma {displaystyle Asubseteq X+P-P} pour certains {displaystyle Xsubseteq A} of cardinality at most {style d'affichage (4ré)^{ré}} . Alors {displaystyle X+P-P} is contained in a generalized arithmetic progression of dimension {style d'affichage |X|+ré} and size at most {style d'affichage 2 ^{|X|}2^{ré}|P|leq 2^{|X|+ré}|2A-2A|leq 2^{|X|+ré}K^{4}|UN|} , complétant la preuve.

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 {style d'affichage G} is a set {displaystyle P+H} for a proper generalized arithmetic progression {style d'affichage P} and a subgroup {style d'affichage H} de {style d'affichage G} . The dimension of this coset progression is defined to be the dimension of {style d'affichage P} , and its size is defined to be the cardinality of the whole set. Green and Ruzsa showed the following: Laisser {style d'affichage A} be a finite set in an abelian group {style d'affichage G} tel que {style d'affichage |A+A|leq K|UN|} . Alors {style d'affichage A} is contained in a coset progression of dimension at most {displaystyle d(K)} and size at most {style d'affichage f(K)|UN|} , où {style d'affichage f(K)} et {displaystyle d(K)} are functions of {style d'affichage K} that are independent of {style d'affichage G} .

Green and Ruzsa provided upper bounds of {displaystyle d(K)=CK^{4}Journal(K+2)} et {style d'affichage f(K)=e^{CK^{4}log ^{2}(K+2)}} for some absolute constant {displaystyle C} .[8] Terence Tao (2010) also generalized Freiman's theorem to solvable groups of bounded derived length.[9] Extending Freiman’s theorem to an arbitrary nonabelian group is still open. Results for {style d'affichage K<2} , when a set has very small doubling, are referred to as Kneser theorems.[10] See also Markov spectrum Plünnecke-Ruzsa inequality Kneser's theorem (combinatorics) References ^ Freiman, G.A. (1964). "Addition of finite sets". Soviet Mathematics. Doklady. 5: 1366–1370. Zbl 0163.29501. ^ Freiman, G. A. (1966). Foundations of a Structural Theory of Set Addition (in Russian). Kazan: Kazan Gos. Ped. Inst. p. 140. Zbl 0203.35305. ^ Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer. ISBN 978-0-387-94655-9. Zbl 0859.11003. p. 252. ^ Ruzsa, Imre Z. (1994). "Generalized arithmetical progressions and sumsets". Acta Mathematica Hungarica. 65 (4): 379–388. doi:10.1007/bf01876039. Zbl 0816.11008. ^ Chang, Mei-Chu (2002). "A polynomial bound in Freiman's theorem". Duke Mathematical Journal. 113 (3): 399–419. CiteSeerX 10.1.1.207.3090. doi:10.1215/s0012-7094-02-11331-3. MR 1909605. ^ Sanders, Tom (2013). "The structure theory of set addition revisited". Bulletin of the American Mathematical Society. 50: 93–127. arXiv:1212.0458. doi:10.1090/S0273-0979-2012-01392-7. ^ Zhao, Yufei. "Graph Theory and Additive Combinatorics" (PDF). ^ Ruzsa, Imre Z.; Green, Ben (2007). "Freiman's theorem in an arbitrary abelian group". Journal of the London Mathematical Society. 75 (1): 163–175. arXiv:math/0505198. doi:10.1112/jlms/jdl021. ^ Tao, Terence (2010). "Freiman's theorem for solvable groups". Contributions to Discrete Mathematics. 5 (2): 137–184. doi:10.11575/cdm.v5i2.62020. ^ Tao, Terence. "An elementary non-commutative Freiman theorem". Terence Tao's blog. Further reading Freiman, G. A. (1999). "Structure theory of set addition". Astérisque. 258: 1–33. Zbl 0958.11008. This article incorporates material from Freiman's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Categories: SumsetsTheorems in number theory