# 塞邁雷迪定理

## 定理敍述

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

${\displaystyle N=N(k,\delta )}$

${\displaystyle r_{k}(N)=o(N).}$

## rk(N) 的具體大小

rk(N) 的確切增長速度仍然未知。目前所知的上下界為

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

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

k = 4 時，和陶哲軒[22][23] 證明了存在 c > 0 使得

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

## 參考資料

1. ^ Erdős, Paul; Turán, Paul. On some sequences of integers (PDF). Journal of the London Mathematical Society. 1936, 11 (4): 261–264. MR 1574918. doi:10.1112/jlms/s1-11.4.261.
2. ^ Roth, Klaus Friedrich. On certain sets of integers. Journal of the London Mathematical Society. 1953, 28 (1): 104–109. MR 0051853. Zbl 0050.04002. doi:10.1112/jlms/s1-28.1.104.
3. ^ Szemerédi, Endre. On sets of integers containing no four elements in arithmetic progression. Acta Math. Acad. Sci. Hung. 1969, 20: 89–104. MR 0245555. Zbl 0175.04301. doi:10.1007/BF01894569.
4. ^ Roth, Klaus Friedrich. Irregularities of sequences relative to arithmetic progressions, IV. Periodica Math. Hungar. 1972, 2: 301–326. MR 0369311. doi:10.1007/BF02018670.
5. ^ Szemerédi, Endre. On sets of integers containing no k elements in arithmetic progression (PDF). Acta Arithmetica. 1975, 27: 199–245. MR 0369312. Zbl 0303.10056. doi:10.4064/aa-27-1-199-245.
6. ^ Erdős, Paul. Some of My Favorite Problems and Results. Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve (编). The Mathematics of Paul Erdős I Second. New York: Springer: 51–70. 2013. ISBN 978-1-4614-7257-5. MR 1425174. doi:10.1007/978-1-4614-7258-2_3.
7. ^ Furstenberg, Hillel. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. D'Analyse Math. 1977, 31: 204–256. MR 0498471. doi:10.1007/BF02813304..
8. ^ Furstenberg, Hillel; Katznelson, Yitzhak; Ornstein, Donald Samuel. The ergodic theoretical proof of Szemerédi’s theorem. Bull. Amer. Math. Soc. 1982, 7 (3): 527–552. MR 0670131. doi:10.1090/S0273-0979-1982-15052-2.
9. Gowers, Timothy. A new proof of Szemerédi's theorem. Geom. Funct. Anal. 2001, 11 (3): 465–588. MR 1844079. doi:10.1007/s00039-001-0332-9.
10. ^ Tao, Terence. The dichotomy between structure and randomness, arithmetic progressions, and the primes. Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; Verdera, Joan (编). International Congress of Mathematicians 1. Zürich: European Mathematical Society: 581–608. 2007. MR 2334204. arXiv:math/0512114. doi:10.4171/022-1/22.
11. O'Bryant, Kevin. Sets of integers that do not contain long arithmetic progressions. Electronic Journal of Combinatorics. 2011, 18 (1). MR 2788676.
12. ^ Behrend, Felix A. On the sets of integers which contain no three terms in arithmetic progression. Proceedings of the National Academy of Sciences. 1946, 23 (12): 331–332. Bibcode:1946PNAS...32..331B. MR 0018694. PMC 1078539. PMID 16588588. Zbl 0060.10302. doi:10.1073/pnas.32.12.331.
13. ^ Rankin, Robert A. Sets of integers containing not more than a given number of terms in arithmetical progression. Proc. Roy. Soc. Edinburgh Sect. A. 1962, 65: 332–344. MR 0142526. Zbl 0104.03705.
14. ^ Elkin, Michael. An improved construction of progression-free sets. Israel Journal of Mathematics. 2011, 184 (1): 93–128. MR 2823971. arXiv:0801.4310. doi:10.1007/s11856-011-0061-1.
15. ^ Green, Ben; Wolf, Julia. A note on Elkin's improvement of Behrend's construction. Chudnovsky, David; Chudnovsky, Gregory (编). Additive number theory. Festschrift in honor of the sixtieth birthday of Melvyn B. Nathanson. New York: Springer: 141–144. 2010. ISBN 978-0-387-37029-3. MR 2744752. arXiv:0810.0732. doi:10.1007/978-0-387-68361-4_9.
16. ^ Bourgain, Jean. On triples in arithmetic progression. Geom. Funct. Anal. 1999, 9 (5): 968–984. MR 1726234. doi:10.1007/s000390050105.
17. ^ Bourgain, Jean. Roth's theorem on progressions revisited. J. Anal. Math. 2008, 104 (1): 155–192. MR 2403433. doi:10.1007/s11854-008-0020-x.
18. ^ Heath-Brown, Roger. Integer sets containing no arithmetic progressions. Journal of the London Mathematical Society. 1987, 35 (3): 385–394. MR 889362. doi:10.1112/jlms/s2-35.3.385.
19. ^ Szemerédi, Endre. Integer sets containing no arithmetic progressions. Acta Math. Hungar. 1990, 56 (1–2): 155–158. MR 1100788. doi:10.1007/BF01903717.
20. ^ Sanders, Tom. On Roth's theorem on progressions. Annals of Mathematics. 2011, 174 (1): 619–636. MR 2811612. arXiv:1011.0104. doi:10.4007/annals.2011.174.1.20.
21. ^ Bloom, Thomas F. A quantitative improvement for Roth's theorem on arithmetic progressions. Journal of the London Mathematical Society. Second Series. 2016, 93 (3): 643–663. MR 3509957. arXiv:1405.5800. doi:10.1112/jlms/jdw010.
22. ^ Green, Ben; Tao, Terence. New bounds for Szemeredi's theorem. II. A new bound for r4(N). Chen, William W. L.; Gowers, Timothy; Halberstam, Heini; Schmidt, Wolfgang; Vaughan, Robert Charles (编). Analytic number theory. Essays in honour of Klaus Roth on the occasion of his 80th birthday. Cambridge: Cambridge University Press: 180–204. 2009. Bibcode:2006math.....10604G. ISBN 978-0-521-51538-2. MR 2508645. Zbl 1158.11007. arXiv:math/0610604.
23. ^ Green, Ben; Tao, Terence. New bounds for Szemerédi's theorem, III: A polylogarithmic bound for r4(N). Mathematika. 2017, 63 (3): 944–1040. MR 3731312. arXiv:1705.01703. doi:10.1112/S0025579317000316.
24. ^ Furstenberg, Hillel; Katznelson, Yitzhak. An ergodic Szemerédi theorem for commuting transformations. Journal d'Analyse Mathématique. 1978, 38 (1): 275–291. MR 531279. doi:10.1007/BF02790016.
25. ^ Gowers, Timothy. Hypergraph regularity and the multidimensional Szemerédi theorem. Annals of Mathematics. 2007, 166 (3): 897–946. MR 2373376. arXiv:0710.3032. doi:10.4007/annals.2007.166.897.
26. ^ Rödl, Vojtěch; Skokan, Jozef. Regularity lemma for k-uniform hypergraphs. Random Structures Algorithms. 2004, 25 (1): 1–42. MR 2069663. doi:10.1002/rsa.20017.
27. ^ Rödl, Vojtěch; Skokan, Jozef. Applications of the regularity lemma for uniform hypergraphs. Random Structures Algorithms. 2006, 28 (2): 180–194. MR 2198496. doi:10.1002/rsa.20108.
28. ^ Nagle, Brendan; Rödl, Vojtěch; Schacht, Mathias. The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms. 2006, 28 (2): 113–179. MR 2198495. doi:10.1002/rsa.20117.
29. ^ Tao, Terence. A variant of the hypergraph removal lemma. Journal of Combinatorial Theory, Series A. 2006, 113 (7): 1257–1280. MR 2259060. arXiv:math/0503572. doi:10.1016/j.jcta.2005.11.006.
30. ^ Bergelson, Vitaly; Leibman, Alexander. Polynomial extensions of van der Waerden's and Szemerédi's theorems. Journal of the American Mathematical Society. 1996, 9 (3): 725–753. MR 1325795. doi:10.1090/S0894-0347-96-00194-4.
31. ^ Furstenberg, Hillel; Katznelson, Yitzhak. A density version of the Hales–Jewett theorem. Journal d'Analyse Mathématique. 1991, 57 (1): 64–119. MR 1191743. doi:10.1007/BF03041066.
32. ^ Wolf, Julia. Finite field models in arithmetic combinatorics—ten years on. Finite Fields and Their Applications. 2015, 32: 233–274. MR 3293412. doi:10.1016/j.ffa.2014.11.003.
33. ^ Yufei Zhao. Origin of my name. [2019-06-17]. （原始内容 (html)存档于2019-06-17）. 叫宇飞
34. ^ Conlon, David; Fox, Jacob; Zhao, Yufei. A relative Szemerédi theorem. Geometric and Functional Analysis. 2015, 25 (3): 733–762. MR 3361771. arXiv:1305.5440. doi:10.1007/s00039-015-0324-9.
35. ^ Zhao, Yufei. An arithmetic transference proof of a relative Szemerédi theorem. Mathematical Proceedings of the Cambridge Philosophical Society. 2014, 156 (2): 255–261. Bibcode:2014MPCPS.156..255Z. MR 3177868. arXiv:1307.4959. doi:10.1017/S0305004113000662.