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

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Number of Parts 
33

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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. 
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Release Date 
2006

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English

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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 (lowcomplexity) 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 GreenTao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemerédis 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  Szemerédis theorem ergodic theory graph theory hypergraph theory arithmetic combinatorics arithmetic progressions prime numbers 
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I they regard start now It's really a very great honor for me to be introducing by Terrence and stuff all of those working there Congress although it had quite a bit about said Terry Yesterday I still find myself but rather a lot But try and that quick
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04:07
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11:57
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And they they use different muscles of for it at the prospect of A similar although For Become very The In any case you can actually use hostel for him us to do something bad The Center for Public Health It was haven't economy he can then innovative and then you get some really useful to get a structure that are often any object and A generic object structure for random become Into a structured and amended And that's really useful because Because Structured optics Amenable to what type of analysis attack using algebra and dormitories and bad things you can attack using public analysis and sold Structure New School for techniques to You understand General Which we would have been accessible subsidy And this This is really Quito along a lot of things for example to the price so The Haganah many examples of such structure of discuss 3 classes But the most movie 1 is especially that That is operator The 6 of a joint over space Then they can can decompose especially in the discrete spectrum which is almost periodic reiterated the view the old South of Macon Compaq and and mixing up which is continuous or continuous But of to the decomposition of the repair decomposition analysis decompose any Any object to a smooth part which is sort of course scales and the hike by What is the position of things and then there was the famous isn't ready regularity photograph of Baker complex You can't work It decomposes The funny thing is you have a the graphs function as you 1 and splitters thing Decomposition But I got this is really of decomposing into a local practitioners of the summed this way Between classes and then you have some some regular grass that between classes And the petition drive structure and the and the having assists from random songs statistics but so these out The restructured often economy And get on and so we have a job functions It's structured done not That Give it to random done it falls through and that so structured You take out And because smaller And now 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27:12
As an example which I think a brighter crown talk about Today Softening An ergodic theory Setting current year because restructure component of All of any donor assistance is force school cactus actor and those because I had no by the new systems which have which arise from group's Newport new groups on dimensional would be groups so popular that so that those are often also were difficult and they're not The other thing and was often great analytic this is 1 of a much larger break But it's proposal bravery general things that's actually a concrete example So For example which Illustrates A lot of these principles wants most of the year The Chris Ford essentially sample it's amazing Watson ready so State We take any subset of images of positive density so you take a 1 per cent of its didn't find but Well it's easy to find that visited me said Dick Sydenham's Munson and for some large you look at the The proportion of The numbers in a it's if this was from below The offer so yet to take what 1 person and 18 % no matter what you do you you Inevitably We are forced to have a Which contains arbitrary Progression amazing it's really quite an amazing is of you get to choose the set him amid given any pattern of example if you want to have to eliminate old city pairs of adjacent numbers And what very easy to choose a set Which avoids What is known as even numbers But you cannot As long as large as the city only what about A lot of no matter how the move around you cannot you cannot destroy Progression he cannot be removed completely West and that's that's really quite an amazing not And this Is buried at 1 1 1 example of 1 of its speeches by seeing how prove that the people proved a selfless on hundreds of ready for 5 contour method 1st proved it again using it and got agreement is very influential in public and Dallas Use of Korea and that there having been accused sent by spoken also another walked out Support on so The most famous is hot add a short list is because of 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34:49
All the people in the region but few Poseidon seal of the quality of so So why we expect of the provisions in the manner that is A little tonight but that But what does what you expect so so in this case A version of course No by structure by the absence of structure by independents coalition Did that to scholarship completely different course progression and then And then we have hybrids commission vote So This is where most sets for the continued a completely random structure that they happen to be Bits of So that we can take to lend structured so We think even numbers structure that Neurotic And wouldn't even numbers to random thoughts of about 2 % The public even numbers at random will the 50 artifacts And then you just 2 random sample even numbers That new demand more structured but above More generally take a cigarette The model of a single circuit of the petition Additionally images destruction pieces and each piece choose band so you might take place 3 % even numbers What is in the odd numbers random And the things that structured progressions and you know that Do I said What's structures progression Utica random subset of structure Matters will continue proportional number Progression to the extent that more general that's The sum which ability problem earlier And so what's structures progressions than anything any random will also have rich as well and so that this because these hybrids progression so in this case the progressions come from a combination of outbreak structure Which which is Initial Canada progression and discoloration Which was refined to Still keep some questions and then the key thing that makes Things work is a structure and depending on the proof that use somebody's professor but we've got structure is different but it's always present And basically The said Think it big dent says that you can always find a large which Which Assad fully surrender but its suitor and set up a structure so example in the because all the hours The a statement a big dance that confined the progression of structures inside except where civil on and what have I done Because Because of what I'm saying is that every that contains high and hybrids as a progression why you accept pressure and be so structured follow phone Economy and because to do so Like this But said It's not doesn't being like some coalition Big Bend The reason is because it college in construction that means Depends proof of the things it some some progression of your cost basis When density This is bigger than normal the other bird doesn't looking for militias of decorum it as always The economy underlying the buckets of tempered versions of these that each proof digs at least call out to talk about what he projects not talk about my own thing green which is a talk about what Dead subset images but Much much faster said But the specific the price of the conjectures about what To aggression Beautiful conjecture according to my conjecture says that A set of Take positive images some of reciprocal diverges then
39:12
The new contained Truong progression so that's much stronger race and it would also like to hear We have no idea of proof that it said At he felt again and another for now what do we have the results for the Anyway But we can use so It is that before And we just result Is not surprising based on the conjectures we have on price we we do believe that the price being randomly after what except for local structure that the affect most or all of the above obvious structures created you expect are randomly members of the size of the country was rejected This The conjecture example would that Leads us to believe private true quite good but it is true as part of its why this is the biggest dick At you Underlies quite a few of the year Police In most remote price so that it and if you ran after should be between aggression but any difficulty but The things we can't verify that we don't know where we still don't know Rented friends we do know 1 thing what are the ones was ousted is a useful we do not I paid from Iast which is a weaker notions Demand essentially the group of prime Except when they obviously small And they became obvious enough smoke coming from local structure Things Some of the wall and the work optional 1 half assembly where huge so the reserve major arts with things of things a bit Said But It was not present in my Knox than we do have sit around and this this is basically a sound and but got off And this is enough do to get some of this conjecture You can get The 1st of the 3 in the price from back and this is done by the corporate 927 But unfortunately it's now well understood that stories is actually Fans Well it it it is for this is a whole haricots around and for years he wrote the list and It's good enough to drop
41:40
3 that even for prisoner of war
41:43
It's not enough examples of event said Which do not have the right number progression of quadratic critics mission example so this is not enough
41:56
Well The increased for Friday or something to talk about How you can boost this To pile pressure from the circle but that does extend but it's not trivial OK so the farms Densities you as for the further funerals of prime But Density and the number so you can't him directly What will they would exploit was that you know the prime spots They Things in a different sense that they don't respect to another set of almost So this Well Almost Kansas had defied the world The way we have to use it as an IQ set of a weight function a measure that But you should think of Almost crimes Numbers virtue prime numbers only state 10 practice at most and those numbers Who was known for time doubt that those guys count because And that they do do behave like we expect The price But they are still after accounted for The obvious reasons but not right and that is Thanks a lot The aimed toward making best starts basically young The primes to meet Extracted the parts of the and expected But since such as the Super tossing that Was accused practice because of example Europe and private Story to lots of fired up to root and then after that what's left of the pride of the runway doing to me steps And It's easy to see what happens meat or what the tumult of but want get more And your I would be doing many many done little changes many many of those who remain a mail small sets Big to greater private party control But if its officers earlier they had been told I'm not screwed event and the 100 than you can still see control on what you have left I will not let me do tricks Essentially can stop What the earlier and still understand exactly what what users but then you don't get plans and we almost saw surely crimes which a cofounder in than and 1 of 100 and have at most 1 wunderkind backed up anyway So the crimes are not case over the last dance In the images Yet The they against almost Prime Besides of almost crimes and a lot of The old define almost Was the most wanted man factors the private density nothing but want to set up effective in almost OK so the way we use that we managed to prove an extension of the Which of course villages of that So not just any death of images agents of all these 2 grand images so But 2 colleges take a representative subset of this case almost tracts of this sort of your focus group which is the 1st of the visitor But even represent the whole images then They need was transfers any fear of proof images should come out As counterpart For this represent a sample is become a part of my bed Dickerson random civilians The things that you're still have of drew on Russia and This is why we get progression of practice because the crimes are dense almost and all of us to abandon by the here and Thank you can see we we'll watch is nothing but a price I could even take
46:07
I did Right they wants in the front and I still which so in in some sense From this I was really a big chief of the reason why I want reasons why the result If desired Versions of crimes as a hopeless you lose price by doing so that we don't actually you very much crimes hit a new low of progression Because we Particularly we have this doesn't settle the question of funds around the world and refuses to avoid a question instead It den subset of something of and that's enough to give all we do have A subsidy program actually tried It plans to act Get bending talk about that and time before updates of the latest took a little bit of a Harley could just as relatives so we used we used this image as 1 ingredients and recombined Chewelah facts so still Again with based on charges that Take dead subset of a random said It could its NEW random itself An intractable what action which now its but that it might not be but nevertheless Anything which you take any death It turns out It It's a large bill which which is to rent It's about a mail receipt of what upset of the whole said Pursuant subset of Dent said But it seems Sample logical back because of injuries Dance concerts Our Siemens sources that had sued Budget And what of privacy shortly but does that anyway If you believe that for some of these the primes the plans contained A large subset which which amounts to random embittered this U.S. something debts and then we have a negligible yourself that says that It gives you an upset What would be a big university very fast Postal container portion what subjected to go assumptions that that Basically pursuance of should contain a fraction of the expected fraction of all 1st big said What have you done Those results not because simmering that then said in a lot of aggression and then This negativity without that I need to a subset of Odense Foster progression that I just told you that There But the prime contained The death of the a registered so that these 2 things ought to be ingredients in an argument but But they fall into the framework of discussing discussed earlier Still the naked abilities Not so difficult it's a just like Schwartz On Yet you Carter Schwartz And you are the main difficulty in proving that set of adaptation of that That is basically what Akashi Schwartz All and defining what to do and that the other difficulty Washington Dallas So Watts So OK so they they have The harder part was structure and destruction Of becomes the economy that Take it and some of us Pursuant to the far almost That's still thought through and then it turns out that actually colleagues with New song structures gone but What They the key of that he said Box structures said that you call it a day And this This summer what what what what drives so the funds will not surrender Because You know Collated bias very structures such as a progression 1 1 of the structures systems like appeared said was a critics said those of sort things that our goal is the dance structures set that up walking 0 and and that's really And This is it that dance is key because accusing instructed that the the that would dent in conclusion which is from the vessel the 2 applies in reduced the band said using the sparse Have Almost the truth because the actual details about meant that the So I just to reward But then I working nowadays 2 years ago And so I just mentioned show you that sets pressure of the density of water progression of price pressure that they don't exactly how many they are listed in the remaining but more Quantitatively Did you ask how progression of lens case other amongst the problems up to some level and And river bank What is the asked can be more precise question was The fact that there are few attitudes A lower back but it's not the shot dead shot so the next question is I thought I had shared you really can't finessed anymore more so than in the in previous year We took the prime on babies man and restructure we don't know That He 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 costefficient 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 highwater 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 Please What did they do call it with them We a job show what what I that but if know sequences don't come from local obstructions expect the tribes Of course example expect all I could do it because critics said much earlier You sector crimes the purported front on the track because of the exception should be the density of access Modules Local structures which led to the about and that can be done by A refined version that although that sort also revealed so that each of these 3 . required to pay his work 3 papers And the social critic of before But But anyway Mike became can now get of progression for And soon hopefully other practices OK The book what war among them think you have