# Szemerédi's theorem

Szemerédi's theorem In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured[1] that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.

Contents 1 Statement 2 History 3 Quantitative bounds 4 Extensions and generalizations 5 See also 6 Notes 7 Further reading 8 External links Statement A subset A of the natural numbers is said to have positive upper density if {displaystyle limsup _{nto infty }{frac {|Acap {1,2,3,dotsc ,n}|}{n}}>0} .

Szemerédi's theorem asserts that a subset of the natural numbers with positive upper density contains infinitely many arithmetic progressions of length k for all positive integers k.

An often-used equivalent finitary version of the theorem states that for every positive integer k and real number {displaystyle delta in (0,1]} , there exists a positive integer {displaystyle N=N(k,delta )} such that every subset of {1, 2, ..., N} of size at least δN contains an arithmetic progression of length k.

Another formulation uses the function rk(N), the size of the largest subset of {1, 2, ..., N} without an arithmetic progression of length k. Szemerédi's theorem is equivalent to the asymptotic bound {displaystyle r_{k}(N)=o(N)} .

That is, rk(N) grows less than linearly with N.

History Van der Waerden's theorem, a precursor of Szemerédi's theorem, was proven in 1927.

The cases k = 1 and k = 2 of Szemerédi's theorem are trivial. The case k = 3, known as Roth's theorem, was established in 1953 by Klaus Roth[2] via an adaptation of the Hardy–Littlewood circle method. Endre Szemerédi[3] proved the case k = 4 through combinatorics. Using an approach similar to the one he used for the case k = 3, Roth[4] gave a second proof for this in 1972.

The general case was settled in 1975, also by Szemerédi,[5] who developed an ingenious and complicated extension of his previous combinatorial argument for k = 4 (called "a masterpiece of combinatorial reasoning" by Erdős[6]). Several other proofs are now known, the most important being those by Hillel Furstenberg[7][8] in 1977, using ergodic theory, and by Timothy Gowers[9] in 2001, using both Fourier analysis and combinatorics. Terence Tao has called the various proofs of Szemerédi's theorem a "Rosetta stone" for connecting disparate fields of mathematics.[10] Quantitative bounds It is an open problem to determine the exact growth rate of rk(N). The best known general bounds are {displaystyle CNexp left(-n2^{(n-1)/2}{sqrt[{n}]{log N}}+{frac {1}{2n}}log log Nright)leq r_{k}(N)leq {frac {N}{(log log N)^{2^{-2^{k+9}}}}},} where {displaystyle n=lceil log krceil } . The lower bound is due to O'Bryant[11] building on the work of Behrend,[12] Rankin,[13] and Elkin.[14][15] The upper bound is due to Gowers.[9] For small k, there are tighter bounds than the general case. When k = 3, Bourgain,[16][17] Heath-Brown,[18] Szemerédi,[19] and Sanders[20] provided increasingly smaller upper bounds. The current best bounds are {displaystyle N2^{-{sqrt {8log N}}}leq r_{3}(N)leq C{frac {(log log N)^{4}}{log N}}N} due to O'Bryant[11] and Bloom[21] respectively.

For k = 4, Green and Tao[22][23] proved that {displaystyle r_{4}(N)leq C{frac {N}{(log N)^{c}}}} for some c > 0.

Extensions and generalizations A multidimensional generalization of Szemerédi's theorem was first proven by Hillel Furstenberg and Yitzhak Katznelson using ergodic theory.[24] Timothy Gowers,[25] Vojtěch Rödl and Jozef Skokan[26][27] with Brendan Nagle, Rödl, and Mathias Schacht,[28] and Terence Tao[29] provided combinatorial proofs.

Alexander Leibman and Vitaly Bergelson[30] generalized Szemerédi's to polynomial progressions: If {displaystyle Asubset mathbb {N} } is a set with positive upper density and {displaystyle p_{1}(n),p_{2}(n),dotsc ,p_{k}(n)} are integer-valued polynomials such that {displaystyle p_{i}(0)=0} , then there are infinitely many {displaystyle u,nin mathbb {Z} } such that {displaystyle u+p_{i}(n)in A} for all {displaystyle 1leq ileq k} . Leibman and Bergelson's result also holds in a multidimensional setting.

The finitary version of Szemerédi's theorem can be generalized to finite additive groups including vector spaces over finite fields.[31] The finite field analog can be used as a model for understanding the theorem in the natural numbers.[32] The problem of obtaining bounds in the k=3 case of Szemerédi's theorem in the vector space {displaystyle mathbb {F} _{3}^{n}} is known as the cap set problem.

The Green–Tao theorem asserts the prime numbers contain arbitrary long arithmetic progressions. It is not implied by Szemerédi's theorem because the primes have density 0 in the natural numbers. As part of their proof, Ben Green and Tao introduced a "relative" Szemerédi theorem which applies to subsets of the integers (even those with 0 density) satisfying certain pseudorandomness conditions. A more general relative Szemerédi theorem has since been given by David Conlon, Jacob Fox, and Yufei Zhao.[33][34] The Erdős conjecture on arithmetic progressions would imply both Szemerédi's theorem and the Green–Tao theorem.

See also Problems involving arithmetic progressions Ergodic Ramsey theory Arithmetic combinatorics Szemerédi regularity lemma Notes ^ Erdős, Paul; Turán, Paul (1936). "On some sequences of integers" (PDF). Journal of the London Mathematical Society. 11 (4): 261–264. doi:10.1112/jlms/s1-11.4.261. MR 1574918. ^ Roth, Klaus Friedrich (1953). "On certain sets of integers". Journal of the London Mathematical Society. 28 (1): 104–109. doi:10.1112/jlms/s1-28.1.104. MR 0051853. Zbl 0050.04002. ^ Szemerédi, Endre (1969). "On sets of integers containing no four elements in arithmetic progression". Acta Mathematica Academiae Scientiarum Hungaricae. 20 (1–2): 89–104. doi:10.1007/BF01894569. MR 0245555. Zbl 0175.04301. ^ Roth, Klaus Friedrich (1972). "Irregularities of sequences relative to arithmetic progressions, IV". Periodica Math. Hungar. 2 (1–4): 301–326. doi:10.1007/BF02018670. MR 0369311. S2CID 126176571. ^ Szemerédi, Endre (1975). "On sets of integers containing no k elements in arithmetic progression" (PDF). Acta Arithmetica. 27: 199–245. doi:10.4064/aa-27-1-199-245. MR 0369312. Zbl 0303.10056. ^ Erdős, Paul (2013). "Some of My Favorite Problems and Results". In Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve (eds.). The Mathematics of Paul Erdős I (Second ed.). New York: Springer. pp. 51–70. doi:10.1007/978-1-4614-7258-2_3. ISBN 978-1-4614-7257-5. MR 1425174. ^ Furstenberg, Hillel (1977). "Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions". J. d'Analyse Math. 31: 204–256. doi:10.1007/BF02813304. MR 0498471. S2CID 120917478.. ^ Furstenberg, Hillel; Katznelson, Yitzhak; Ornstein, Donald Samuel (1982). "The ergodic theoretical proof of Szemerédi's theorem". Bull. Amer. Math. Soc. 7 (3): 527–552. doi:10.1090/S0273-0979-1982-15052-2. MR 0670131. ^ Jump up to: a b Gowers, Timothy (2001). "A new proof of Szemerédi's theorem". Geom. Funct. Anal. 11 (3): 465–588. doi:10.1007/s00039-001-0332-9. MR 1844079. S2CID 124324198. ^ Tao, Terence (2007). "The dichotomy between structure and randomness, arithmetic progressions, and the primes". In Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; Verdera, Joan (eds.). Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006. International Congress of Mathematicians. Vol. 1. Zürich: European Mathematical Society. pp. 581–608. arXiv:math/0512114. doi:10.4171/022-1/22. ISBN 978-3-03719-022-7. MR 2334204. ^ Jump up to: a b O'Bryant, Kevin (2011). "Sets of integers that do not contain long arithmetic progressions". Electronic Journal of Combinatorics. 18 (1). doi:10.37236/546. MR 2788676. ^ Behrend, Felix A. (1946). "On the sets of integers which contain no three terms in arithmetic progression". Proceedings of the National Academy of Sciences. 32 (12): 331–332. Bibcode:1946PNAS...32..331B. doi:10.1073/pnas.32.12.331. MR 0018694. PMC 1078964. PMID 16578230. Zbl 0060.10302. ^ Rankin, Robert A. (1962). "Sets of integers containing not more than a given number of terms in arithmetical progression". Proc. Roy. Soc. Edinburgh Sect. A. 65: 332–344. MR 0142526. Zbl 0104.03705. ^ Elkin, Michael (2011). "An improved construction of progression-free sets". Israel Journal of Mathematics. 184 (1): 93–128. arXiv:0801.4310. doi:10.1007/s11856-011-0061-1. MR 2823971. ^ Green, Ben; Wolf, Julia (2010). "A note on Elkin's improvement of Behrend's construction". In Chudnovsky, David; Chudnovsky, Gregory (eds.). Additive Number Theory. Additive number theory. Festschrift in honor of the sixtieth birthday of Melvyn B. Nathanson. New York: Springer. pp. 141–144. arXiv:0810.0732. doi:10.1007/978-0-387-68361-4_9. ISBN 978-0-387-37029-3. MR 2744752. S2CID 10475217. ^ Bourgain, Jean (1999). "On triples in arithmetic progression". Geom. Funct. Anal. 9 (5): 968–984. doi:10.1007/s000390050105. MR 1726234. S2CID 392820. ^ Bourgain, Jean (2008). "Roth's theorem on progressions revisited". J. Anal. Math. 104 (1): 155–192. doi:10.1007/s11854-008-0020-x. MR 2403433. S2CID 16985451. ^ Heath-Brown, Roger (1987). "Integer sets containing no arithmetic progressions". Journal of the London Mathematical Society. 35 (3): 385–394. doi:10.1112/jlms/s2-35.3.385. MR 0889362. ^ Szemerédi, Endre (1990). "Integer sets containing no arithmetic progressions". Acta Mathematica Hungarica. 56 (1–2): 155–158. doi:10.1007/BF01903717. MR 1100788. ^ Sanders, Tom (2011). "On Roth's theorem on progressions". Annals of Mathematics. 174 (1): 619–636. arXiv:1011.0104. doi:10.4007/annals.2011.174.1.20. MR 2811612. S2CID 53331882. ^ Bloom, Thomas F. (2016). "A quantitative improvement for Roth's theorem on arithmetic progressions". Journal of the London Mathematical Society. Second Series. 93 (3): 643–663. arXiv:1405.5800. doi:10.1112/jlms/jdw010. MR 3509957. S2CID 27536138. ^ Green, Ben; Tao, Terence (2009). "New bounds for Szemeredi's theorem. II. A new bound for r4(N)". In Chen, William W. L.; Gowers, Timothy; Halberstam, Heini; Schmidt, Wolfgang; Vaughan, Robert Charles (eds.). New bounds for Szemeredi's theorem, II: A new bound for r_4(N). Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press. pp. 180–204. arXiv:math/0610604. Bibcode:2006math.....10604G. ISBN 978-0-521-51538-2. MR 2508645. Zbl 1158.11007. ^ Green, Ben; Tao, Terence (2017). "New bounds for Szemerédi's theorem, III: A polylogarithmic bound for r4(N)". Mathematika. 63 (3): 944–1040. arXiv:1705.01703. doi:10.1112/S0025579317000316. MR 3731312. ^ Furstenberg, Hillel; Katznelson, Yitzhak (1978). "An ergodic Szemerédi theorem for commuting transformations". Journal d'Analyse Mathématique. 38 (1): 275–291. doi:10.1007/BF02790016. MR 0531279. S2CID 123386017. ^ Gowers, Timothy (2007). "Hypergraph regularity and the multidimensional Szemerédi theorem". Annals of Mathematics. 166 (3): 897–946. arXiv:0710.3032. doi:10.4007/annals.2007.166.897. MR 2373376. S2CID 56118006. ^ Rödl, Vojtěch; Skokan, Jozef (2004). "Regularity lemma for k-uniform hypergraphs". Random Structures Algorithms. 25 (1): 1–42. doi:10.1002/rsa.20017. MR 2069663. ^ Rödl, Vojtěch; Skokan, Jozef (2006). "Applications of the regularity lemma for uniform hypergraphs" (PDF). Random Structures Algorithms. 28 (2): 180–194. doi:10.1002/rsa.20108. MR 2198496. ^ Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias (2006). "The counting lemma for regular k-uniform hypergraphs". Random Structures Algorithms. 28 (2): 113–179. doi:10.1002/rsa.20117. MR 2198495. ^ Tao, Terence (2006). "A variant of the hypergraph removal lemma". Journal of Combinatorial Theory, Series A. 113 (7): 1257–1280. arXiv:math/0503572. doi:10.1016/j.jcta.2005.11.006. MR 2259060. ^ Bergelson, Vitaly; Leibman, Alexander (1996). "Polynomial extensions of van der Waerden's and Szemerédi's theorems". Journal of the American Mathematical Society. 9 (3): 725–753. doi:10.1090/S0894-0347-96-00194-4. MR 1325795. ^ Furstenberg, Hillel; Katznelson, Yitzhak (1991). "A density version of the Hales–Jewett theorem". Journal d'Analyse Mathématique. 57 (1): 64–119. doi:10.1007/BF03041066. MR 1191743. S2CID 123036744. ^ Wolf, Julia (2015). "Finite field models in arithmetic combinatorics—ten years on". Finite Fields and Their Applications. 32: 233–274. doi:10.1016/j.ffa.2014.11.003. MR 3293412. ^ Conlon, David; Fox, Jacob; Zhao, Yufei (2015). "A relative Szemerédi theorem". Geometric and Functional Analysis. 25 (3): 733–762. arXiv:1305.5440. doi:10.1007/s00039-015-0324-9. MR 3361771. S2CID 14398869. ^ Zhao, Yufei (2014). "An arithmetic transference proof of a relative Szemerédi theorem". Mathematical Proceedings of the Cambridge Philosophical Society. 156 (2): 255–261. arXiv:1307.4959. Bibcode:2014MPCPS.156..255Z. doi:10.1017/S0305004113000662. MR 3177868. S2CID 119673319. Further reading Tao, Terence (2007). "The ergodic and combinatorial approaches to Szemerédi's theorem". In Granville, Andrew; Nathanson, Melvyn B.; Solymosi, József (eds.). Additive Combinatorics. CRM Proceedings & Lecture Notes. Vol. 43. Providence, RI: American Mathematical Society. pp. 145–193. arXiv:math/0604456. Bibcode:2006math......4456T. ISBN 978-0-8218-4351-2. MR 2359471. Zbl 1159.11005. External links PlanetMath source for initial version of this page Announcement by Ben Green and Terence Tao – the preprint is available at math.NT/0404188 Discussion of Szemerédi's theorem (part 1 of 5) Ben Green and Terence Tao: Szemerédi's theorem on Scholarpedia Weisstein, Eric W. "SzemeredisTheorem". MathWorld. Grime, James; Hodge, David (2012). "6,000,000: Endre Szemerédi wins the Abel Prize". Numberphile. Brady Haran. Categories: Additive combinatoricsRamsey theoryTheorems in combinatoricsTheorems in number theory

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