塞邁雷迪定理

在算術組合學（英語：arithmetic combinatorics）中，塞邁雷迪定理（英語：Szemerédi's theorem）是個關於自然數集子集中的等差數列的結論。1936年，艾狄胥·帕爾和圖蘭·帕爾猜想^[1]：若整數集 A 具有正的自然密度，則對任意的正整數 k, 都可以在 A 中找出一個 k 項的等差數列。匈牙利數學家塞邁雷迪·安德烈於1975年證明了此結論。

定理敍述

若自然數集的子集 A 滿足

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

則稱 A 具有正的上密度。塞邁雷迪定理斷言，若自然數集的一個子集具有正的上密度，則對任意的正整數 k, 該子集都包含無窮多個長為 k 的（公差不為 0) 的等差數列。

定理另有一個等價的有限性敍述（即不牽涉「無窮」的）：對任意的正整數 k 和實數 $\delta \in (0,1],$ 都存在正整數

N=N(k,\delta )

使得 {1, 2, ..., N} 的每個至少 δN 元的子集，都包含長為 k 的等差數列。

定義 r_k(N) 為 {1, 2, ..., N} 的子集當中，不包含長為 k 的等差數列的最大子集的大小。塞邁雷迪定理給出漸近上界

r_{k}(N)=o(N).

換言之，r_k(N) 隨 N 的增長慢於線性。

歷史

范德瓦爾登定理是塞邁雷迪定理的先驅，其於 1927 年獲證。

塞邁雷迪定理中， k = 1 和 k = 2 的情況顯然成立。 k = 3 的結論關乎薩萊姆-斯賓塞集（英語：Salem-Spencer set）（不包含 3 項等差數列的整數子集）的大小。1953 年，克勞斯·羅特^[2] 利用類似哈代-李特爾伍德圓法的方法，證明了 k = 3 的結論。1969 年，塞邁雷迪·安德烈^[3]利用組合學方法證明了 k = 4 的情況。在 1972 年，羅特^[4]利用類似自己證明 k = 3 的情況的方法，給出了 k = 4 的情況的另證。

塞邁雷迪於 1975 年給出了適用於所有 k 的證明。^[5] 他巧妙地擴展了自己先前對 k = 4 的情況的組合論證，艾狄胥稱該證明為「組合學推理的傑作」("a masterpiece of combinatorial reasoning")^[6]. 定理亦有若干其他證明，較重要的有：1977 年希勒爾·菲爾斯滕貝格利用遍歷理論給出的證明^[7]^[8]，以及 2001 年高爾斯結合傅里葉分析和組合數學給出的證明^[9]。陶哲軒稱塞邁雷迪定理的眾多證明為「羅塞塔石碑」，因為它們連結了幾個乍看之下迥異的數學分支。^[10]

r_k(N) 的具體大小

r_k(N) 的確切增長速度仍然未知。目前所知的上下界為

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}}}}},

其中 $n=\lceil \log k\rceil .$ 歐布萊恩（Kevin O'Bryant) 證出了上述的下界^[11] ，其證明建基於貝哈連德（Felix Behrend)^[12]、羅伯特·藍欽（英語：Robert Alexander Rankin）^[13]，以及埃爾金（Michael Elkin) ^[14]^[15]先前的成果。上界由高爾斯證出。^[9]

對於較小的 k, 可以找到比起一般情況更緊的上下界。當 k = 3 時，布爾甘^[16]^[17]、希斯-布朗^[18]、塞邁雷迪^[19]，以及湯姆·桑德斯^[20]依次給出了愈來愈好的下界。目前所知的上下界為

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

兩邊的界限分別由歐布萊恩^[11] 和布魯姆（Thomas Bloom) ^[21] 給出。

當 k = 4 時，本·格林（英語：Ben Green (mathematician)）和陶哲軒^[22]^[23] 證明了存在 c > 0 使得

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

推廣

希勒爾·菲爾斯滕貝格和伊扎克·卡茨內勒松（英語：Yitzhak Katznelson）利用遍歷理論整明了塞邁雷迪定理的高維推廣。^[24] 高爾斯^[25]、沃伊傑赫·勒德爾（英語：Vojtěch Rödl）和約澤夫·斯科肯 (Jozef Skokan) ^[26]^[27] ，以及布蘭登·納格 (Brendan Nagle)、勒德爾和馬蒂亞斯·沙赫特（英語：Mathias Schacht）^[28] ，和陶哲軒^[29]給出了各自的組合學證明。

亞歷山大·萊布門 (Alexander Leibman) 和維塔利·別爾格爾松（英語：Vitaly Bergelson）^[30] 給出定理對多項式列的推廣：若 $A\subset \mathbb {N}$ 的上密度為正，且 $p_{1}(n),p_{2}(n),\dotsc ,p_{k}(n)$ 為滿足 $p_{i}(0)=0$ 的整值多項式（英語：Integer-valued polynomial），則存在無窮多組 $u,n\in \mathbb {Z}$ 使得對 $1\leq i\leq k$ 都有 $u+p_{i}(n)\in A.$ 萊布門和別爾格爾松的結果同樣適用於高維的情況。

塞邁雷迪定理的有限性版本可推廣至有限的加法群上，例如有限域上的向量空間。^[31] 定理在有限域上的類比，是有助理解原定理（在正整數集上）的模型。^[32] 所謂封頂集（英語：Cap set）問題，就是要找出向量空間 $\mathbb {F} _{3}^{n}$ 所包含的無 3 項等差數列的最大子集的大小，即塞邁雷迪定理當 k = 3 時的界限。

格林-陶定理斷言，存在任意長的質數等差數列。此結論不能由塞邁雷迪定理推出，因為質數集的密度為 0. 本·格林（英語：Ben Green (mathematician)）和陶哲軒在其證明中引入了「相對性」（英語：relative) 的塞邁雷迪定理，該定理適用於任意具特定偽隨機性的整數子集（允許密度為 0)。大衛·康倫（英語：David Conlon），雅各布·福克斯（英語：Jacob Fox）和趙宇飛^[33] (Yufei Zhao)其後亦給出了適用於更一般情況的相對性塞邁雷迪定理。^[34]^[35]

埃爾德什等差數列猜想（斷言倒數和發散的集合，必有任意長的等差數列）可以推出塞邁雷迪定理和格林-陶定理。

參見

與等差數列有關的問題（英語：Problems involving arithmetic progressions）
遍歷拉姆齊理論（英語：Ergodic Ramsey theory）
算術組合學（英語：arithmetic combinatorics）
塞邁雷迪正則性引理

參考資料

^ Erdős, Paul; Turán, Paul. On some sequences of integers (PDF). Journal of the London Mathematical Society. 1936, 11 (4): 261–264 [2019-06-17]. MR 1574918. doi:10.1112/jlms/s1-11.4.261. （原始內容存檔 (PDF)於2020-07-23）.
^ 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.
^ 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.
^ Roth, Klaus Friedrich. Irregularities of sequences relative to arithmetic progressions, IV. Periodica Math. Hungar. 1972, 2: 301–326. MR 0369311. doi:10.1007/BF02018670.
^ Szemerédi, Endre. On sets of integers containing no k elements in arithmetic progression (PDF). Acta Arithmetica. 1975, 27: 199–245 [2019-06-17]. MR 0369312. Zbl 0303.10056. doi:10.4064/aa-27-1-199-245. （原始內容存檔 (PDF)於2020-12-10）.
^ Erdős, Paul. Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve , 編. 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. |chapter=被忽略 (幫助)
^ 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. .
^ 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.
^ ^9.0 ^9.1 Gowers, Timothy. A new proof of Szemerédi's theorem. Geom. Funct. Anal. 2001, 11 (3): 465–588 [2019-06-17]. MR 1844079. doi:10.1007/s00039-001-0332-9. （原始內容存檔於2020-09-26）.
^ Tao, Terence. Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; Verdera, Joan , 編. International Congress of Mathematicians 1. Zürich: European Mathematical Society: 581–608. 2007. MR 2334204. arXiv:math/0512114 . doi:10.4171/022-1/22. |chapter=被忽略 (幫助)
^ ^11.0 ^11.1 O'Bryant, Kevin. Sets of integers that do not contain long arithmetic progressions. Electronic Journal of Combinatorics. 2011, 18 (1) [2019-06-17]. MR 2788676. （原始內容存檔於2018-05-03）.
^ 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.
^ 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.
^ 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.
^ Green, Ben; Wolf, Julia. Chudnovsky, David; Chudnovsky, Gregory , 編. 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. |chapter=被忽略 (幫助)
^ Bourgain, Jean. On triples in arithmetic progression. Geom. Funct. Anal. 1999, 9 (5): 968–984. MR 1726234. doi:10.1007/s000390050105.
^ 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.
^ Heath-Brown, Roger. Integer sets containing no arithmetic progressions. Journal of the London Mathematical Society. 1987, 35 (3): 385–394. MR 0889362. doi:10.1112/jlms/s2-35.3.385.
^ Szemerédi, Endre. Integer sets containing no arithmetic progressions. Acta Math. Hungar. 1990, 56 (1–2): 155–158. MR 1100788. doi:10.1007/BF01903717.
^ 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.
^ 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.
^ Green, Ben; Tao, Terence. Chen, William W. L.; Gowers, Timothy; Halberstam, Heini; Schmidt, Wolfgang; Vaughan, Robert Charles , 編. 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 . |chapter=被忽略 (幫助)
^ 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.
^ Furstenberg, Hillel; Katznelson, Yitzhak. An ergodic Szemerédi theorem for commuting transformations. Journal d'Analyse Mathématique. 1978, 38 (1): 275–291. MR 0531279. doi:10.1007/BF02790016.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ Yufei Zhao. Origin of my name. [2019-06-17]. （原始內容 (html)存檔於2019-06-17）. 叫宇飛
^ 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.
^ 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.

延申閱讀

Tao, Terence. Granville, Andrew; Nathanson, Melvyn B.; Solymosi, József , 編. Additive Combinatorics. CRM Proceedings & Lecture Notes 43. Providence, RI: American Mathematical Society: 145–193. 2007. Bibcode:2006math......4456T. ISBN 978-0-8218-4351-2. MR 2359471. Zbl 1159.11005. arXiv:math/0604456 . |chapter=被忽略 (幫助)

外部鏈結

Announcement by Ben Green and Terence Tao （頁面存檔備份，存於互聯網檔案館） – the preprint is available at math.NT/0404188
Discussion of Szemerédi's theorem (part 1 of 5) （頁面存檔備份，存於互聯網檔案館）
Ben Green and Terence Tao: Szemerédi's theorem （頁面存檔備份，存於互聯網檔案館） on Scholarpedia
埃里克·韋斯坦因. SzemeredisTheorem. MathWorld.
Grime, James; Hodge, David. 6,000,000: Endre Szemerédi wins the Abel Prize. Numberphile. 布雷迪·哈蘭. 2012 [2019-06-17]. （原始內容存檔於2014-01-09）.

[erdos_turan-1] Erdős, Paul; Turán, Paul. On some sequences of integers (PDF). Journal of the London Mathematical Society. 1936, 11 (4): 261–264 [2019-06-17]. MR 1574918. doi:10.1112/jlms/s1-11.4.261. （原始內容存檔 (PDF)於2020-07-23）.

[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 [2019-06-17]. MR 0369312. Zbl 0303.10056. doi:10.4064/aa-27-1-199-245. （原始內容存檔 (PDF)於2020-12-10）.

[6] Erdős, Paul. Graham, Ronald L.; Nešetřil, Jaroslav; Butler, Steve , 編. 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. |chapter=被忽略 (幫助)

[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.

[gowers-9] 9.0 ^9.1 Gowers, Timothy. A new proof of Szemerédi's theorem. Geom. Funct. Anal. 2001, 11 (3): 465–588 [2019-06-17]. MR 1844079. doi:10.1007/s00039-001-0332-9. （原始內容存檔於2020-09-26）.

[10] Tao, Terence. Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; Verdera, Joan , 編. International Congress of Mathematicians 1. Zürich: European Mathematical Society: 581–608. 2007. MR 2334204. arXiv:math/0512114 . doi:10.4171/022-1/22. |chapter=被忽略 (幫助)

[obryant-11] 11.0 ^11.1 O'Bryant, Kevin. Sets of integers that do not contain long arithmetic progressions. Electronic Journal of Combinatorics. 2011, 18 (1) [2019-06-17]. MR 2788676. （原始內容存檔於2018-05-03）.

[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. Chudnovsky, David; Chudnovsky, Gregory , 編. 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. |chapter=被忽略 (幫助)

[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 0889362. 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. Chen, William W. L.; Gowers, Timothy; Halberstam, Heini; Schmidt, Wolfgang; Vaughan, Robert Charles , 編. 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 . |chapter=被忽略 (幫助)

[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 0531279. 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.

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定理敍述

歷史

rk(N) 的具體大小

推廣

參見

參考資料

延申閱讀

外部鏈結

r_k(N) 的具體大小