# Szemerédi's theorem

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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, 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 while also introducing what is now called the Gowers norm. 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] Sanders,[20] and Bloom[21] established progressively smaller upper bounds, and Bloom and Sisask then proved the first bound that broke the so-called "logarithmic barrier".[22] The current best bounds are

${\displaystyle N2^{-{\sqrt {8\log N}}}\leq r_{3}(N)\leq Ne^{-c(\log N)^{1/9}}}$, for some constant ${\displaystyle c>0}$,

respectively due to O'Bryant,[11] and Bloom and Sisask[23] (the latter built upon the breakthrough result of Kelley and Meka,[24] who obtained the same upper bound, with "1/9" replaced by "1/12").

For k = 4, Green and Tao[25][26] proved that

${\displaystyle r_{4}(N)\leq C{\frac {N}{(\log N)^{c}}}}$

For k=5 in 2023 and k≥5 in 2024 Leng, Sah and Sawhney proved in preprints [27][28][29] that:

${\displaystyle r_{k}(N)\leq CN\exp(-(\log \log N)^{c})}$

## Extensions and generalizations

A multidimensional generalization of Szemerédi's theorem was first proven by Hillel Furstenberg and Yitzhak Katznelson using ergodic theory.[30] Timothy Gowers,[31] Vojtěch Rödl and Jozef Skokan[32][33] with Brendan Nagle, Rödl, and Mathias Schacht,[34] and Terence Tao[35] provided combinatorial proofs.

Alexander Leibman and Vitaly Bergelson[36] 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 ${\displaystyle p_{i}(0)=0}$, 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.[37] The finite field analog can be used as a model for understanding the theorem in the natural numbers.[38] 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 arbitrarily 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.[39][40]

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. doi:10.1112/jlms/s1-11.4.261. MR 1574918.
2. ^ 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.
3. ^
4. ^ 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.
5. ^ 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.
6. ^ 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.
7. ^
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. doi:10.1090/S0273-0979-1982-15052-2. MR 0670131.
9. ^ 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.
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 (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.
11. ^ a b O'Bryant, Kevin (2011). "Sets of integers that do not contain long arithmetic progressions". Electronic Journal of Combinatorics. 18 (1). arXiv:0811.3057. doi:10.37236/546. MR 2788676.
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13. ^ Rankin, Robert A. (1962). "Sets of integers containing not more than a given number of terms in arithmetical progression". Proc. R. Soc. Edinburgh Sect. A. 65: 332–344. MR 0142526. Zbl 0104.03705.
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15. ^ 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.
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19. ^ Szemerédi, Endre (1990). "Integer sets containing no arithmetic progressions". Acta Mathematica Hungarica. 56 (1–2): 155–158. doi:10.1007/BF01903717. MR 1100788.
20. ^ 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.
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22. ^ Bloom, Thomas F.; Sisask, Olof (2020). "Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions". arXiv:2007.03528v2 [math.NT].
23. ^ Bloom, Thomas F.; Sisask, Olof (2023). "An improvement to the Kelley-Meka bounds on three-term arithmetic progressions". arXiv:2309.02353v1 [math.NT].
24. ^ Kelley, Zander; Meka, Raghu (2023). "Strong Bounds for 3-Progressions". arXiv:2302.05537v5 [math.NT].
25. ^ 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.
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29. ^ Sloman, Leila (2024-08-05). "Grad Students Find Inevitable Patterns in Big Sets of Numbers". Quanta Magazine. Retrieved 2024-08-07.
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