# 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.

## Statement

A subset A of the natural numbers is said to have positive upper density if

${\displaystyle \limsup _{n\to \infty }{\frac {|A\cap \{1,2,3,\dotsc ,n\}|}{n}}>0}$.

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 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

${\displaystyle CN\exp \left(-n2^{(n-1)/2}{\sqrt[{n}]{\log N}}+{\frac {1}{2n}}\log \log N\right)\leq r_{k}(N)\leq {\frac {N}{(\log \log N)^{2^{-2^{k+9}}}}},}$

where ${\displaystyle n=\lceil \log k\rceil }$. 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

${\displaystyle N2^{-{\sqrt {8\log N}}}\leq r_{3}(N)\leq C{\frac {(\log \log N)^{4}}{\log N}}N}$

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 Furstenberg and Katznelson using ergodic theory.[24] Gowers,[25] Rödl–Skokan[26][27] with Nagle–Rödl–Schacht,[28] and Tao[29] provided combinatorial proofs.

Leibman and Bergelson[30] generalized Szemerédi's to polynomial progressions: If ${\displaystyle A\subset \mathbb {N} }$ 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 ${\displaystyle p_{i}(k)=0}$, then there are infinitely many ${\displaystyle u,n\in \mathbb {Z} }$ such that ${\displaystyle u+p_{i}(n)\in A}$ for all ${\displaystyle 1\leq i\leq k}$. 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 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, 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 Conlon, Fox, and Zhao.[33][34]

The Erdős conjecture on arithmetic progressions would imply both Szemerédi's theorem and the Green–Tao theorem.

## Notes

1. ^ Erdős, Paul; Turán, Paul (1936). "On some sequences of integers" (PDF). Journal of the London Mathematical Society. 11 (4): 261–264. MR 1574918. doi:10.1112/jlms/s1-11.4.261.
2. ^ Roth, Klaus Friedrich (1953). "On certain sets of integers". Journal of the London Mathematical Society. 28 (1): 104–109. MR 0051853. Zbl 0050.04002. doi:10.1112/jlms/s1-28.1.104.
3. ^ Szemerédi, Endre (1969). "On sets of integers containing no four elements in arithmetic progression". Acta Math. Acad. Sci. Hung. 20: 89–104. MR 0245555. Zbl 0175.04301. doi:10.1007/BF01894569.
4. ^ Roth, Klaus Friedrich (1972). "Irregularities of sequences relative to arithmetic progressions, IV". Periodica Math. Hungar. 2: 301–326. MR 0369311. doi:10.1007/BF02018670.
5. ^ Szemerédi, Endre (1975). "On sets of integers containing no k elements in arithmetic progression" (PDF). Acta Arithmetica. 27: 199–245. MR 0369312. Zbl 0303.10056.
6. ^ Erdős, Paul (2013). "Some of My Favorite Problems and Results". In Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve. The Mathematics of Paul Erdős I (Second ed.). New York: Springer. pp. 51–70. ISBN 978-1-4614-7257-5. MR 1425174. doi:10.1007/978-1-4614-7258-2_3.
7. ^ Furstenberg, Hillel (1977). "Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions". J. D'Analyse Math. 31: 204–256. MR 0498471. doi:10.1007/BF02813304..
8. ^ Furstenberg, Hillel; Katznelson, Yitzhak; Ornstein, Donald Samuel (1982). "The ergodic theoretical proof of Szemerédi’s theorem". Bull. Amer. Math. Soc. 7 (3): 527–552. MR 0670131. doi:10.1090/S0273-0979-1982-15052-2.
9. ^ a b Gowers, Timothy (2001). "A new proof of Szemerédi's theorem". Geom. Funct. Anal. 11 (3): 465–588. MR 1844079. doi:10.1007/s00039-001-0332-9.
10. ^ 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. International Congress of Mathematicians. 1. Zürich: European Mathematical Society. pp. 581–608. MR 2334204. arXiv:. doi:10.4171/022-1/22.
11. ^ a b O'Bryant, Kevin (2011). "Sets of integers that do not contain long arithmetic progressions". Electronic Journal of Combinatorics. 18 (1). MR 2788676.
12. ^ Behrend, Felix A. (1946). "On the sets of integers which contain no three terms in arithmetic progression". Proceedings of the National Academy of Sciences. 23 (12): 331–332. MR 0018694. Zbl 0060.10302. doi:10.1073/pnas.32.12.331.
13. ^ 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.
14. ^ Elkin, Michael (2011). "An improved construction of progression-free sets". Israel Journal of Mathematics. 184 (1): 93–128. MR 2823971. doi:10.1007/s11856-011-0061-1.
15. ^ Green, Ben; Wolf, Julia (2010). "A note on Elkin's improvement of Behrend's construction". In Chudnovsky, David; Chudnovsky, Gregory. Additive number theory. Festschrift in honor of the sixtieth birthday of Melvyn B. Nathanson. New York: Springer. pp. 141–144. ISBN 978-0-387-37029-3. MR 2744752. doi:10.1007/978-0-387-68361-4_9.
16. ^ Bourgain, Jean (1999). "On triples in arithmetic progression". Geom. Funct. Anal. 9 (5): 968–984. MR 1726234. doi:10.1007/s000390050105.
17. ^ Bourgain, Jean (2008). "Roth's theorem on progressions revisited". J. Anal. Math. 104 (1): 155–192. MR 2403433. doi:10.1007/s11854-008-0020-x.
18. ^ Heath-Brown, Roger (1987). "Integer sets containing no arithmetic progressions". Journal of the London Mathematical Society. 35 (3): 385–394. MR 889362. doi:10.1112/jlms/s2-35.3.385.
19. ^ Szemerédi, Endre (1990). "Integer sets containing no arithmetic progressions". Acta Math. Hungar. 56 (1-2): 155–158. MR 1100788. doi:10.1007/BF01903717.
20. ^ Sanders, Tom (2011). "On Roth's theorem on progressions". Annals of Mathematics. 174 (1): 619–636. MR 2811612. doi:10.4007/annals.2011.174.1.20.
21. ^ 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. MR 3509957. arXiv:. doi:10.1112/jlms/jdw010.
22. ^ 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. Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press. pp. 180–204. ISBN 978-0-521-51538-2. MR 2508645. Zbl 1158.11007. arXiv:.
23. ^ Green, Ben; Tao, Terence (2017). "New bounds for Szemerédi's theorem, III: A polylogarithmic bound for r4(N)". arXiv:.
24. ^ Furstenberg, Hillel; Katznelson, Yitzhak (1978). "An ergodic Szemerédi theorem for commuting transformations". Journal d'Analyse Mathématique. 38 (1): 275–291. MR 531279. doi:10.1007/BF02790016.
25. ^ Gowers, Timothy (2007). "Hypergraph regularity and the multidimensional Szemerédi theorem". Ann. of Math. 166 (3): 897–946. MR 2373376. doi:10.4007/annals.2007.166.897.
26. ^ Rödl, Vojtěch; Skokan, Jozef (2004). "Regularity lemma for k-uniform hypergraphs". Random Structures Algorithms. 25 (1): 1–42. MR 2069663. doi:10.1002/rsa.20017.
27. ^ Rödl, Vojtěch; Skokan, Jozef (2006). "Applications of the regularity lemma for uniform hypergraphs". Random Structures Algorithms. 28 (2): 180–194. MR 2198496. doi:10.1002/rsa.20108.
28. ^ Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias (2006). "The counting lemma for regular k-uniform hypergraphs". Random Structures Algorithms. 28 (2): 113–179. MR 2198495. doi:10.1002/rsa.20117.
29. ^ Tao, Terence (2006). "A variant of the hypergraph removal lemma". Journal of Combinatorial Theory. Series A. 113 (7): 1257–1280. MR 2259060. doi:10.1016/j.jcta.2005.11.006.
30. ^ 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. MR 1325795. doi:10.1090/S0894-0347-96-00194-4.
31. ^ Furstenberg, Hillel; Katznelson, Yitzhak (1991). "A density version of the Hales–Jewett theorem". Journal d'Analyse Mathématique. 57 (1): 64–119. MR 1191743. doi:10.1007/BF03041066.
32. ^ Wolf, Julia (2015). "Finite field models in arithmetic combinatorics—ten years on". Finite Fields and Their Applications. 32: 233–274. MR 3293412. doi:10.1016/j.ffa.2014.11.003.
33. ^ Conlon, David; Fox, Jacob; Zhao, Yufei (2015). "A relative Szemerédi theorem". Geometric and Functional Analysis. 25 (3): 733–762. MR 3361771. arXiv:. doi:10.1007/s00039-015-0324-9.
34. ^ Zhao, Yufei (2014). "An arithmetic transference proof of a relative Szemerédi theorem". Mathematical Proceedings of the Cambridge Philosophical Society. 156 (2): 255–261. MR 3177868. doi:10.1017/S0305004113000662.