|
|
| Line 1: |
Line 1: |
| __NOTOC__
| | Gabrielle Straub is what you can call me although it's not the quite a few feminine of names. Fish keeping is what I provide every week. Managing people is my month job now. My house is now with regard to South Carolina. Go to my website to find out more: http://circuspartypanama.com<br><br>Also visit my page; [http://circuspartypanama.com clash of clans hack download no survey no password] |
| '''Szemerédi's theorem''' is a result In [[arithmetic combinatorics]], concerning [[arithmetic progression]]s in subset of the integers.
| |
| In 1936, [[Paul Erdős|Erdős]] and [[Paul Turán|Turán]] conjectured<ref name="erdos turan">{{citation|authorlink1=Paul Erdős|first1=Paul|last1=Erdős|authorlink2=Paul Turán|first2=Paul|last2=Turán|title=On some sequences of integers|journal=[[Journal of the London Mathematical Society]]|volume=11|issue=4|year=1936|pages=261–264|url=http://www.renyi.hu/~p_erdos/1936-05.pdf|doi=10.1112/jlms/s1-11.4.261}}.</ref> that every set of integers ''A'' with positive [[natural density]] contains a ''k'' term [[arithmetic progression]] for every ''k''. This conjecture, which became Szemerédi's theorem, generalizes the statement of [[van der Waerden's theorem]].
| |
| | |
| ==History==
| |
| The cases ''k'' = 1 and ''k'' = 2 are trivial. The case ''k'' = 3 was established in 1953 by [[Klaus Roth]]<ref>{{citation|authorlink=Klaus Friedrich Roth|first=Klaus Friedrich|last=Roth|title=On certain sets of integers, I|journal=[[Journal of the London Mathematical Society]]|volume=28|pages=104–109|year=1953|id=Zbl 0050.04002, {{MathSciNet|id=0051853}}|doi=10.1112/jlms/s1-28.1.104}}.</ref> via an adaptation of the [[Hardy–Littlewood circle method]]. The case ''k'' = 4 was established in 1969 by [[Endre Szemerédi]]<ref>{{citation|authorlink=Endre Szemerédi|first=Endre|last=Szemerédi|title=On sets of integers containing no four elements in arithmetic progression|journal=Acta Math. Acad. Sci. Hung.|volume=20|pages=89–104|year=1969|id=Zbl 0175.04301, {{MathSciNet|id=0245555}}|doi=10.1007/BF01894569}}.</ref> via a combinatorial method. Using an approach similar to the one he used for the case ''k'' = 3, Roth<ref>{{citation|authorlink=Klaus Friedrich Roth|first=Klaus Friedrich|last=Roth|title=Irregularities of sequences relative to arithmetic progressions, IV|journal=Periodica Math. Hungar.|volume=2|pages=301–326|year=1972|id={{MathSciNet|id=0369311}}|doi=10.1007/BF02018670}}.</ref> gave a second proof for this in 1972.
| |
| | |
| Finally, the case of general ''k'' was settled in 1975, also by Szemerédi,<ref>{{citation|authorlink=Endre Szemerédi|first=Endre|last=Szemerédi|title=On sets of integers containing no ''k'' elements in arithmetic progression|journal=Acta Arithmetica|volume=27|pages=199–245|year=1975|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27132.pdf|id=Zbl 0303.10056, {{MathSciNet|id=0369312}}}}.</ref> via an ingenious and complicated extension of the previous combinatorial argument (called "a masterpiece of combinatorial reasoning" by [[Ronald Graham|R. L. Graham]]). Several further proofs are now known, the most important amongst them being those by [[Hillel Furstenberg]]<ref>{{citation|authorlink=Hillel Furstenberg|first=Hillel|last=Furstenberg|title=Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions|journal=J. D'Analyse Math.|volume=31|pages=204–256|year=1977|id={{MathSciNet|id=0498471}}|doi=10.1007/BF02813304}}.</ref> in 1977, using [[ergodic theory]], and by [[Timothy Gowers]]<ref name="gowers">{{citation|authorlink=Timothy Gowers|first=Timothy|last=Gowers|title=A new proof of Szemerédi's theorem|journal=Geom. Funct. Anal.|volume=11|issue=3|pages=465–588|url=http://www.dpmms.cam.ac.uk/~wtg10/sz898.dvi|year=2001|id={{MathSciNet|id=1844079}}|doi=10.1007/s00039-001-0332-9}}.</ref> in 2001, using both [[Fourier analysis]] and [[combinatorics]].
| |
| | |
| ==Finitary version==
| |
| Let ''k'' be a positive integer and let 0 < δ ≤ 1/2. A [[finitary]] version of the theorem states that there exists a positive integer
| |
| | |
| :<math>N = N(k,\delta)\,</math>
| |
| | |
| such that every subset of {1, 2, ..., ''N''} of size at least δ''N'' contains an arithmetic progression of length ''k''.
| |
| The best known bounds for ''N''(''k'', δ) are
| |
| | |
| :<math>C^{\log(1/\delta)^{k-1}} \leq N(k,\delta) \leq 2^{2^{\delta^{-2^{2^{k+9}}}}}</math>
| |
| | |
| with ''C'' > 1. The lower bound is due to [[Felix A. Behrend|Behrend]]<ref>{{citation|authorlink=Felix A. Behrend|first=Felix A.|last=Behrend|title=On the sets of integers which contain no three in arithmetic progression|journal=[[Proceedings of the National Academy of Sciences]]|volume=23|issue=12|pages=331–332|year=1946|id=Zbl 0060.10302|doi=10.1073/pnas.32.12.331}}.</ref> (for ''k'' = 3) and [[Robert Alexander Rankin|Rankin]],<ref>{{citation|authorlink=Robert A. Rankin|first=Robert A.|last=Rankin|title=Sets of integers containing not more than a given number of terms in arithmetical progression|journal=Proc. Roy. Soc. Edinburgh Sect. A|volume=65|pages=332–344|year=1962|id=Zbl 0104.03705, {{MathSciNet|id=0142526}}}}.</ref> and the upper bound is due to Gowers.<ref name="gowers"/>
| |
| When ''k'' = 3 better upper bounds are known; [[Jean Bourgain|Bourgain]]<ref>{{citation|authorlink=Jean Bourgain|first=Jean|last=Bourgain|title=On triples in arithmetic progression|journal=Geom. Func. Anal.|volume=9|issue=5|year=1999|pages=968–984|id={{MathSciNet|id=1726234}}|doi=10.1007/s000390050105}}</ref> has proved that
| |
| | |
| :<math>N(3,\delta) \leq C^{\delta^{-2}\log(1/\delta)},</math>
| |
| | |
| This was improved by [[Tom Sanders (mathematician)|Sanders]]<ref>{{citation|authorlink=Tom Sanders (mathematician)|first=Tom|last=Sanders|title=On Roth's theorem on progressions|journal=Annals of Mathematics|volume=174|issue=1|year=2011|pages=619–636|id={{MathSciNet|id=2811612}}|doi=10.4007/annals.2011.174.1.20}}</ref> to
| |
| | |
| :<math>N(3,\delta) \leq C^{\delta^{-1}(\log(1/\delta))^5}</math>
| |
| | |
| ==See also==
| |
| * [[Problems involving arithmetic progressions]]
| |
| * [[Ergodic Ramsey theory]]
| |
| * [[Arithmetic combinatorics]]
| |
| * [[Szemerédi regularity lemma]]
| |
| * [[Green–Tao theorem]]
| |
| * [[Erdős conjecture on arithmetic progressions]]
| |
| | |
| ==Notes==
| |
| {{reflist|colwidth=30em}}
| |
| | |
| ==Further reading==
| |
| * {{Citation | first=Terence | last=Tao | authorlink=Terence Tao | chapter=The ergodic and combinatorial approaches to Szemerédi's theorem | pages=145–193 | editor1-first=Andrew | editor1-last=Granville | editor2-first=Melvyn B. | editor2-last=Nathanson| editor3-first=József | editor3-last=Solymosi | title=Additive Combinatorics | series= CRM Proceedings & Lecture Notes | volume=43 | publisher=[[American Mathematical Society]] | year=2007 | isbn=978-0-8218-4351-2 | zbl=1159.11005 }}
| |
| | |
| ==External links==
| |
| * [http://planetmath.org/encyclopedia/SzemeredisTheorem.html PlanetMath source for initial version of this page]
| |
| * [http://www.math.ucla.edu/~tao/whatsnew.html Announcement by Ben Green and Terence Tao] – the preprint is available at [http://front.math.ucdavis.edu/math.NT/0404188 math.NT/0404188]
| |
| * [http://in-theory.blogspot.com/2006/06/szemeredis-theorem.html Discussion of Szemerédi's theorem (part 1 of 5)]
| |
| * Ben Green and Terence Tao: [http://www.scholarpedia.org/article/Szemeredi%27s_Theorem Szemerédi's theorem] on [[Scholarpedia]]
| |
| * {{MathWorld|SzemeredisTheorem|SzemeredisTheorem}}
| |
| * {{cite web|last=Grime|first=James|title=6,000,000: Endre Szemerédi wins the Abel Prize|url=http://www.numberphile.com/videos/abel_prize.html|work=Numberphile|year=2012|publisher=[[Brady Haran]]|coauthors=Hodge, David}}
| |
| | |
| {{DEFAULTSORT:Szemeredi's theorem}}
| |
| [[Category:Ramsey theory]]
| |
| [[Category:Theorems in combinatorics]]
| |
| [[Category:Theorems in number theory]]
| |
Gabrielle Straub is what you can call me although it's not the quite a few feminine of names. Fish keeping is what I provide every week. Managing people is my month job now. My house is now with regard to South Carolina. Go to my website to find out more: http://circuspartypanama.com
Also visit my page; clash of clans hack download no survey no password