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

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The dichotomy between structure and randomness, arithmetic progressions, and the primes
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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 even so you have to make adjustments Will want to obstructions would 3 modified girls You put all those working together this product Which is the number 1 . 3 something I think it will be looking beanstalk and so on A victory in the precise count of 0 how fast it goes depends on removing officers that we know 0 even on unconditional no hypotheses the way people use these hiding Commander and the U.S. didn't start and the hundreds of comedy shows you that if you want to establish a result you yet a corporate crimes The local history what's going on They jocks circle And all you need to do is to show that On my Knox that A partner local you want you want a who is the political crimes More than you can 1 neglect them and then your local trips and local churches of work would a product of the Well The info analysis we destruction of its completely class that Characters exponential your not surrender of years surrender call it with a character But needs of a large work and We need to do is 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 hiding the the and a previous so we we we generalize So common that replaced Food aid with a more Hi of the dose uniformity And basically the key point is that You wanna show that crimes After the meeting Local obstructions which you always should be kept out of stronger than for And Not not strong enough to do things account twins In prime Sokol but did not account these motor premiered at the reversal of for the Fabrizio autocratic was replaced Point and step aside to practice on And So The point is that ensure analysis structure of its characters The high-water case structure of its more complicated than they have recently declassified at least For the year And a getaway for progress in the war Former 1 groups have what no sequences coming from a cautionary importantly group and that is There is a natural organization all characters flows coming home from work Taurus and and is still want to have classification which is not trivial but we need to do is show that The prime don't 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