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The dichotomy between structure and randomness, arithmetic progressions, and the primes


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Title The dichotomy between structure and randomness, arithmetic progressions, and the primes
Title of Series International Congress of Mathematicians, Madrid 2006
Number of Parts 33
Author Tao, Terence
License CC Attribution 3.0 Germany:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
DOI 10.5446/15977
Publisher Instituto de Ciencias Matemáticas (ICMAT)
Release Date 2006
Language English

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Subject Area Mathematics
Abstract A famous theorem of Szemerédi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemerédi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green–Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.
Keywords Szemerédi’s theorem
ergodic theory
graph theory
hypergraph theory
arithmetic combinatorics
arithmetic progressions
prime numbers
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case Progression but depending what history and you'll find a different number progression and getting what exact count of progressed he have to know How many times are found and That sort of Linux focus Can't say that I get an ingenious way to start using the year of the other leaders of the I mentioned relied mostly on the 1st 2 games of the 1st 3 of his negligible ability the economy and the structure of the wheels of his rigidity and cost-efficient and those Up to give you deeper information As a What more quantitative information So the prototype of this is this Was and hope that that Your account progression 3 . 5 must end The answer is actually quit it over So instead of the normal person week period the presidency wearable again So Square log tube and the 3 us and we want after that but Local The crimes of honor with 1 exception which which increases the number of all the Ferguson We could order because 2 numbers Progression expected orders well And what got lucky we can't that 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to contemplate between students structures that you have to show the prime don't they don't call a minor heart attack Show that the crimes don't call it was to And 2 guys and there's a merciful across the Chautauqua head You do that End What does that and you get this thing which is right so it's much stronger than significant pressure over 3 harmony So More recently than I would to do the same thing for other crimes which started provision for I hope eventually we will look look said we get Andy Near the primacy in which Which involves Except for those that only 1 parameter They have a go at least 2 and apprenticed averaging Neglect We see parts so it wasn't so bad Really annoying caviar could we could do things are different from conjecture legal but conditional on things But we can't do that we can Can account for the cubes How iPads the crimes of the civil service of dimension of war again so the lovable What the description of this so and this is much more an extension of the 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