5/6 Nilsequences

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5/6 Nilsequences
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Classical Fourier analysis has found many uses in additive number theory. However, while it is well-adapted to some pro - blems, it is unable to handle others. For example, if one has a set A, and one wishes to know how many 3-term arithmetic progressions are contained in A, then Fourier analysis is useful, but if one wishes to count 4-term progressions then it is not. For this, and other, problems the more general notion of a nilsequence is required. NIlsequences are a kind of «higher order character» forming the basis of what is becoming known as «higher-order Fourier analysis». The talks will be about this theory.
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and my best if I might a and we can do in be the past 2 and my he didn't want to talk about something related to somebody's mind but before I get started on that I would like to say a few words about the inverse served for the garrison which is something I mentioned several times In the my statement of the facts about from what was 1 of the gap is still there "quotation mark which is why function of Class 1 which correlates with we call some positive constants and I spent in fact whole out during the detailed formulation of which requires you to put some by the complexity with bans on complexity all 4 so I mention this many times I it is very difficult for a long time to prove but I think he's got some ideas of how the proof goes and with 1 thing in particular the we have neglected to mention so far which is the case cake was actually very straightforward it's going to make a few remarks on the purse the classical for and when I did some classical for analysis In my 1st lecture I got the normalization from Rome no no savings left again road games nobody to correct me down and the point here anyway is that there's a formula the into all of 2 the powerful well it all actually I'm normalized and actually also made this an additional constantly coming from the fact that I'm not working group but basically and this can be expanded so it's constantly accused time intervals from 1 of the average value 4 so this this is a former that's very similar to the former assured you for solution for copies of X Y and X plus wife awaited by you can establish by orthogonal authorization careful the relations we have to combine with the possibility if possible the old to normal this summer acts of X square is to go from 1 all of thailand's where you compare these 2 facts will result was quite easily said he on under the assumption that the garrison all of that is at least of having 2 and stuff like that implies that the I'm all for it transformed into expanded by Constance go for a but on the other hand disease stop the square time failed to compared to use the possibility to get it is impossible to give a clear the fact that flight "quotation mark give can equals to theories gives a very specific type With kind "quotation mark of 1 class 1 but
what happened here is what this is no sequence with a vertical frequency I introduced the notion of vertical frequency the before and while ministry true with the assistance of the strong statement that kind of frequency and you just get up free from the inverse stated plotted the composition of high into no sequences with frequency some of 2 steps get concatenated together inside anyway the point is that this is a relatively straightforward and in the general case is done by induction on for the general case induction on using the crucial relations which I think I'm pretty sure I mentioned before using the fact that the the UK or for the UK plus 1 or 2 the power to suffocate plus 1 is an average of UK norms of derivatives that it's the average age all multiplicative the power to the well here don't have access to mathematics the slight technical technical inaccuracies in what I've written here to do with the fact that this range the fact that the want when it's not quite but that's would be literature if I was working in the same group in which better since the fact that I had it get UK plus 1 known is the stealth than that implies that many of the derivatives that have a large UK you get to the fall back down I didn't work constantly worsened to the cape that for form many users and then want to have a good working conclude that derivative correlates with no sequence of class on 1 this implies that the average acts of about half of exports H Bomb ,comma sub H by by calls of I was relatively straightforward the difficulties is then integrating this assertion about derivatives to get an assertion about function itself to get some kind of feel for what the task might involved and we now need to win integrates the statement my concern is that started with were just a quarter quarter the president-elect is heated to the next then what could I take kind to be inside the I can take I believe that I can take heart stoppage affects to eat too big to fit the age acts and actually would just be it wouldn't have mattered 1 constant the on so you can see in of this is by no means the weight varies with age is by no means all this ferries and 1 would expect something like the same phenomenon to be true generally at least for the top auditor whatever that means all of this no sequence ,comma so we hit upon the idea all establishing trying to stop some kind of linear behavior because of that so we need to we're going to have to do that wisely this has another property which is that it's symmetric in it turns out that you have to establish some kind of a version of that property as well also this comes a bit later down the line symmetry so what you got to gather evidence that this kind of Asian is behaving as it should it really is a derivative of something that's of course 1 great so how is the stuff while the argument is is a beautiful argument due to guarantee that gives you compliment you wouldn't be changes simplification of cows argument which reduces some of the magical give way that should be absolute values around them but let me just ignore that and in fact if you just twist the whole thing by appropriate phase you can ignore so endangered northern and from I
just get the average age and sex all of next to explore the stage times ,comma of major acts banned and make a change of variables that may just be doubly sure which ones yeah but at the cost of a sequence of spaces really the 1st change and what you get implies that the average and and all of them ,comma on Monday all right responded and again I've been a little bit vague about the Rangers these things going on but that's a purely technical matters so now we can apply because shots inequalities in fact the way in which case short applied very similar to the way it's applied in many other parts of the trial gave any details in any particular case many and not give any details that have but I think it's sort of clear what you do so you can supply Keisha shots 1st in the end variable I like to to delete have because and then square what remains and then outlined in the end variable plus a new dummy variable that came out of the square and that allows me to delete these ones as well I to applications of coach she turns to the implies that there is an empire and price of corn and money and time of the appropriate dashed expressions it is time to below and when you consider this it In a statement of income that has disappeared that's because I want to prove some statements about head high varies with age unfortunately it's not the 1st class looking statement there I can make it a bit more pleasant weather another substitution we remember what set was certain each 1 you In addition to this expression and become the average 8 1 8 2 3 4 the average of color of 8 1 call because of H 2 and was 1 of the 4 kinds of H 4 and stage 1 mistake for its big 1 actually this constraints on these 8 1 2 each for the satisfying so men may not have to look any better than what I had to fall but let me show you know what happens in a specific case so that we were in the case of this is the case equals to so when we did reduction the derivatives would correlate with a linear because that's the I think that case kind of each of his EU fees that and if you substitute in here while you get a geometric series but what you get so the phase will be just the the 8 1 should be symbolic
feature-rich 1 minus the courage to minus the rich 3 plus the for is the phase that you all are averaging and when he that GPU is negligible unless there is a very great places to produce calculations thank what you get is many more than constant times and Q quarter of old age 1 each to H 3 and H 4 with which call portables 1 2 equals aged 3 4 schools supporters and with a very similar property feature the 1 minus the courage to the courageous 3 plus the search for is basically bounded by 1 of but the symbols here contains dependent on them I'm actually I'm not entirely sure got my science I'll stick with that this looks a little bit more suggestive had agree to suggesting that the but it really does look a little bit like a homomorphism have a genuine homomorphism take relation like this and send it to the surface suggests that's not a master like home office well good because I wanted to show precisely that the variation of the originates but unfortunately it's not well actually fortunate because otherwise the whole theory would be long this does not imply that January genuine homomorphism so there are some examples of maps like this which satisfy these 2 properties but which is not a homomorphism would be a map of the form of the charade is something like the brackets the stage when the fractional part so why would that satisfy the property well restricted to the Saturday each for which this fractional part is less than one-tenth let's this is a genuine this will be because the fractional parties additive for such fractional parts but it's not a Gallegly additive function said this is not a genuine home office now we are just examples is not good enough attend the conference that examples so I think that the idea of amounts that can be halted for the please 1 the state of precise and we have the jury which would equal the sum of a few brackets terms and then you have to allow tolerance as well because that's the assumption that when Diaz and this is just this is for a large fraction of and so this In fact of moralism the work you have to go on this topic and it uses tools from Madison ,comma topics which I don't go to the elections that uses what's now called the about some goes and Prime Minister some facts from the German numbers while I guess they are inside the profits from the this is quite a nontrivial factor I have to the sketch proved more direct deduction of this form of from this which I haven't fully worked out the details of it seems as if this is such a natural statement 1 should have to go through all of that machinery I think that this is quite and it was something that may strike you as if you remembered the 1st couple of lectures it that this is not the 1st time we've seen brackets objects to the brackets should remind us of something we've seen before bracket my fault heart and if you but derivatives of kind Asian and is buried in this bracket where then it is in fact the derivative of something coming from the heart and groups if you wish establish a certain symmetry so we can pretend that we can construct sequence come from all this which could cause that kind of year of this brackets but the same can't always do that you need to establish a day at the cemetery statement as well all kinds of age at the ancient acts and not to reinstate what the statement of work a sketch of how the provisions for the 3 would go and while it's complicated but the other thing I want to emphasize budgets 1 reason I'm giving lectures and writing the Associated is that this is just the wrong in that the Eisenberg Group which is for natural Nobleton and group associated with the problem is
constructed in a totally ad hoc manner by 1st findings in brackets expressions slightly and said the lectures somehow wanted never really be working with bracketing expressions if you can help me no present objects that are more natural just construct by inspection the Heisenberg and no sequence on the heightened by group which gave rise to the so it doesn't seem natural and what I would like somebody to find a more natural argument is a different approach to all of this by Secord the head of this 1 of the 3 by 3 yeah so you you could make each 1 of us from Cobb 3 3 but you can prove actually directly that citizens no surprise you could prove just by looking at I things some simple conversations with cost to the algebra that you can make every example from lies up some small errors so you guys both you know in advance that they could be enough the the what the isolation of a product of the house and exists to dimension to the you could also realize this is coming from a larger dimension cost to prove it is not a product of her but this amount of flexibility unique representation 2 so the we have to tolerate this roughly what happens here's a one-dimensional bracket office and this will be on the office of maybe Hall whenever the fractional part of this is less than the 1 between months a quarter to quarter and if you have the brackets there should be more than a fraction To to the monastery of the time so you certainly need to let the growing like 1 of adults you have to let agree this is a big difference between the class to call 1 of the last 1 could always just use the objects on so ,comma said and here you need to to let dimensions so I was going to say that there is another approach to these bisected Ian Cameron 2nd which is quite different and in a sense is more natural because you see a militant group is more conceptual work but it's equally difficult in them in the details the doesn't provide and I think both arguments and not the right argument for the proof of the investor the the only but this is the kind of kind of a dream of mine 1 could find inappropriate for the analysis of these costs to objects but it would have to be very exhausted because you don't have your camera monetary is not guaranteed to begin with so you can't just decompose L 2 I mean you know that you are going to consider many of the achievement ,comma you have to allow arbitrarily large dimensions here but I don't have no Bill ideas of what such a theory could look like I do believe there's something to be said it's different this seductive approach this was just train for the year so no 1 respects in which the analysis here was very straightforward is that I got just summer geometric progression in this part of the argument and for the U I would get to do that so many have an average over objects which are themselves class too and to value the unity the theory that I mentioned last time but he quit diffusion of sequences and get it's much more difficult to get still harder for the flight normal sitting down did prove a statement for for all of the UK norms and want to stay where he proved Gallas In the state from 1998 to this very difficult but they can I think it's if you 3 I don't think it's better the fall in the absence says the same thing opposed in the UK known as of why and get at correlating with object on all of the interval 1 or 2 end but rather some local correlation if you from that way function correlate with local can just be the constant weakened politicians want to into progression all right equally so in all like a with a 2 to the beach and don't the letter at least an hour power some small depending on delves into the delta other defendants here is awesome quite reasonable but I will mention that so they're much smaller than the original incident started with but there's still of size tending to infinity with this is definitely not the saying the violence equal and the average of progression abounded away from so it's much less information just very local correlation now it seems as if this is a strong pheromone that you get local correlations with just a constant function but actually this can be really rather it achieved as consequence all which has been calling there is a sequence is actually constant "quotation mark fought appear to be call but I should said I mean I'm not playing in the garage result is a trivial consequences some result that has been established by cedar myself run Our resources requires all of the techniques essentially guarantees and then some more user gives it back so if you don't care about Bonds while
for anybody use Donna little better balance in number theory leave as an exercise for you why square calls on the the Net maybe precise about what we might we have and the politicians each at least and some small constant as such the money from the variation of the of the and square the hands of P on of the the people the and it is less than a 10th of the month the constant so that's the kind of statement is true more generally for sequences but in this case you could establish that buys standard techniques involving both quality and and so on it's not easy but it's classical a case of the questions on what assets from yes gas has to papers this the paper which I feel that everybody should read masterful paper on the 4th and aggression where he proves basically this for these 3 actually state he stated quite like this so they follow in he has some additional quadratic for him but then if you apply this observation you write a sketch on some smart aggression local constant and then he has a much longer papers so that it is only 23 pages and were truly remarkable piece of work then along the pages of a 29 pages is to the general case and that he runs into difficulties that are somewhat parallels and difficulties we have with the uniform His motivations understand that not yet again motivation was summary so I stated that the 1st lecture and let me just remind you that summary states that it is a subset of 1 of the size of a time then take progression provided lodging so Dallas drew inspiration from an argument of class role in the case of cake was 3 and this is called the density increment strategy the idea that the increase is to prove a dichotomy "quotation mark the form I followed 2 things happen contains many oppression all you have a distinct so you have the compost to stop aggression on which the density of pay is increased there's some progression a big because and some constant the alpha constant the danced on the bit is out of office ,comma so constable depend on so if you establish such a dichotomy you can do and it's a it's written argument start with if it contains many progressions than your finished otherwise you participate in and then you apply the same argument so that contains many digressions although the further progression and then he repeats Have you can you repeat abounded number of times in terms of office before you end up with something ridiculous because the density is always the most what so you plan to intuitive sense of application of passengers right that's what the depend on with that of 1 or 2 calls I e-mails where dependents could come in the answer is that actually there is a 3rd of slightly trivial you can't really have a settlement of a dichotomy I'm going to try so this a sort of trivial option which is is that and was too small it's time leave might so the strategy and it to establish this dichotomy what you share is that if this fails and then is characteristic function has a large garrison all and then there was case and I should care about him 1 and I will just mention what that's changes the 2nd Amendment that implies that you have a density increment by the local investors so that's how that's the direction but that request he's almost concede that the statement that comes of the local investors remained average of the characteristic function of a much of while if its average on a progression is large that's precisely the same thing as saying that the density of a progression of and was a trick to get rid of the the absolute values on which you need to do any questions so is the 1st step that mentioned that go good job on many giving 6 lectures and moralist destroyed all of the chalk tiny bits yeah that gays for what I call the generalize from London recalled general from if I write team
took care of that 1 can be won by Karadzic with function then backed by the gap go on there is a 1 and as I mentioned before this itself is you use uses immediately to count the number of progressions and said that if you slightly ,comma but so will apply this respect the characteristic function as a plan that would of course be what we could about function and then despite the multitude as the number of digressions 40 at 1 2 1 it's going to be offered care plus a bunch of terms the final 1 which will be along with works in the least 1 can and each of these terms is bounded by the garrison of so this will equal of the can the speed of the game at this point the dichotomy clay the left side of the number of progression if it's small well then this time here would have to counsel with this time after the case not small it's not 0 so the anyway have no digressions and as if the the garrison Council the the questions on this applies so that's all I'm going to say about the guy was proof what I want to finish today talk about a more exotic variances and register that I mentioned in the 1st lecture they are used much more comprehensive so let me state of Armageddon so this is a myself time 2010 but it's not fair that had previously been established an ergodic theory although the conclusion quantitatively it's it's weak but analogous but the host so it's a kind of excessive strengthening of summaries of the 3 important aggression that is a subset of 1 of the size of France and there is a common difference the nor is such that it has How the can modestly lower and can aggression all right ,comma and many of his 304 was also true K 1 or 2 but I think I can leave exercise an example of British shows that its falls the case for war show with that this really used structure no group the released sketch it's going to give you some idea the the 1st period and this is something that I have not and this is something that we have give slightly strange names called the arithmetical regularity I know stated precisely because it's not particularly straightforward but what it states is that in all the 3 function although it rebounded function from 1 to enter the complexes can be compared right at the close of play uniform but the basic idea is that the acts of sequence 1 what I should be more disciplined about whether scale it is just a dummy variable but some have used the the occasion to length the progression in which case an interested in Gallas came on as 1 of the disposes of sequence of class at fixed 1 I can decompose functions no sequence of Class S plus some things that really should don't concern me me too much of it to cut occasionally bit noise and that this is small out small and little else to which is made you can always ignore this 1 it really small In the dark U.S. plus 1 now that the it's small senators minutes small independent Alan fantasy so random plus or minus 1 function could be listed here the invited Athens around and plus minus 1 function this decomposition would like plus cluster but this can be ignored if you're doing things like counting progressions it is equal to the case of progression began came on as 1 of the the best in woods which is to stay sequences came this means I can ignore when 1 way to think of I mean of course that statement be convicted that if all I care about is counting on progression then I can assume that every function is unknown chemist the
price the regularity involves an iterated applications of it inside what's called an energy increment arguments just keep applying invest and everything and eventually he spent decomposition like that because something called the energy has gone up every time you do that much like the density document that can only happen upon a number of times you know I think this is and In the Class World afford this is a bit like decomposing of functions into its big for a confession
and the rest of the 4th confession supporters of applied that regularity led to my given that I want once again find quite a bit more advantages this year for discussion let's just assume that 1 of them is equal to the characteristic function of some nice said all of he said where is the most just like an open book and agents that not very complicated so maybe I should say that I'm going to prove to them that as stated in the case calls for and in the case whose full I applied whose because then I will be more than the guarantees free and its terms that a small in the EU's 3 that can be ignored when counting Fortin aggression so this is an oversimplification of the I mean whilst I will have a class to know and group may not be the Heisenberg and maybe told me that the function that I take on Eisenberg is just the characteristic function said not even a bona fide no sequence because I find them to use functions on June 1 ,comma the discussion of the just a sequence that really mind what the topic as Alpha Beta highly dependent 1 alpha-beta highly independent Q which means by what I said last nite that the the pier that is very secretive highly from it's not completely Kurdish freedom because it's a finite set it's pretty good distributed in the government chief then the last time what we give them 2 steps forward and progressions and In the wake of course we abide by my assumption it she is equal to the number of Fort aggressions involving but said last time that the distribution of this topic appear appearance of the last 2 he was 3 that does not depend on secret is to beat to the fore a equidistant inside something called hope so they could distribute in a group of cheese for Game for calls trust or for and I didn't give the general definition of what my group is to gave 1 example but which has is going to be found export 6 1 6 2 6 no injuries for so in Heidelberg to the fore which satisfies there'd be abelian parts are enforced and oppression projections to talk to you altar and the core components of the old 1 the next the extent I that let people know about the G 1 1 1 you got that I'm rising part G 8 and G essential parts are not enough progression that they do satisfy constraints it's the constraint collapse monasteries 6 1 plus 3 cents next to minus the frequency the example I given the hope Etruscan group lost time which was for for the real's all with a funny filtration we only saw all this relationship I'm Eisenberg is a kind of a billion bits and essential a Bill projection of billions I think it's true what so when you could kind
of depicts the situation like this is great inaccurate and someone that a billion all the Heisenberg no manifold you get this is suspected Gama'a monetary most of the comments which is isomorphic to almost square and began McGee itself since above and I certainly do not have the ability to draw the factories that may just be a three-dimensional objects that it's flooded above the aspires to be circuits so far above a typical for aggression would have like the the image of fat so the image of 4th of oppression under common I would like to 410 depression down here and then in the vertical fighters will satisfy the traditional relations this relation that went like a four-time progression in the fibers that wasn't quite make sense but it will satisfy constraint know what I haven't done yet is pick my common difference the theory that I'm trying to prove is about a specific date I'm going to change my duty and in fact to argument work choose many tea with this property but which don't move too much in the horizontal directions after D & B in the Alps pretty much the same Thurmond alpha and beta what occurred in the pollen and on pain so far but they fought without comment the end of the 10 plus 2 2 and 3 for aggression with such a common different then it said it doesn't do anything horizontally done the store is the alpha very small basically just the full points down here I will be very very close together and then be there images will be 4 points out here that it essentially the same certified so here on the events of the past 2 pavement plus 3 years essentially constant horizontally .period the quality of the above and I'm working set S & Eisenberg groups of the sunset what lies at the end of the sequel to pass and fired so the number for the professions for which the common difference the it is essentially constant dance that is essentially the number of solutions to this equation these vertical flight so let me try and run so the average number of forts and progression this is a specialty special "quotation mark the integral role solutions to that knowledge monasteries said 1 plus 3 cents -minus said 3 of the characteristic function of this slice St
I said no 1 so that's just another way of dozens of formalization of what I've been saying If you have a progression that essentially constant in the base it's the same thing as a solution to the linear equation in the vertical fighters who just said they call progression which arise because the base and it's it's equal to the 2nd thing that I just the point is that this is has a special property special positivity property and the actually right the office ,comma Square 1 of the star 1 of month on 3 St where the and by Casey shots that's at least average at Of the integral the money the but this is just precisely measure last season path for the average all the measure in the vertical fiber arresting force and by holders of quality that's at least the measure power for so St onto slices of setting and so forth power offers again at least for now and that is just the set of values that for which he analyzing happy about could attributes just 4 so that's why it's not an easy argument but that's the explanation of of why this theory that I wrote the truth and the key thing that I want to emphasize is that there is a certain positivity property that is associated equation if you have a set of measures .period the measure solutions that equations leased out is powerful and if you try and do OK fine well you everything that's similar except that instead he would not have we have to step falls and the question of fibers is no longer on this 1 it's the next row Pascal's triangle of BZ Myners 1 plus 6 cents to 1 for the 3 plus 4 equals and that equation does not have the same positivity properties so you certainly can't use this trick of writing as a convolution square and indeed there are examples of subsets of the circle of measure Delta for which the measure solutions to this equation is less than those of the 5 I'm really not that so that some languages 1st constructed with too few solutions to that and then defines his AS set of return times of of acute To that set OK so that'll be it for today and then tomorrow I'm going to talk about how to apply these results to the problem all please don't mind actually would have a dramatic effect yeah I think you will see friends yeah it is a little bit like the barrel construction 3 professions such as constructing sets with no solution to this ,comma similar past it's very related to the phenomenon that you can have set of the offer with less than half of the 100 times and squared 310 correction I a we will not use this tomorrow the but it is the only people here know about graph theory that sort of combinatorics is quite analogous to somebody regularity land which can be thought of as a way of decomposing all the tree into structured pieces plus a kind of students sued around park I should more accurately this is analogous to the hypergraph version of statement becomes with very very bad and that's the problem and the reason is that for it to be useful you need to have a great function instead sees this they will be James great functions which
would typically be the exponential function and then you can make space smaller than 1 over that background function at all the complexity of if you don't do that you don't get to know all this when you
so that's why it's kind of complicated reinstated properly it's like the traditional regularity level for crosses about recorded some regular and actually do something called strong regularity which this is equivalent which is best seen as the statement was more like the actually find there is no case deterrent regularity stated precisely like growth the structure here you use the it's not provide argument by arguments can be seen as an effective isolation of projection and helpless so what you really want to do is decomposed got L 2 and sitting inside they have something like L 2 of Tremont Calif but quantitative version of version that you want to project we don't know how else to Duval County have the the space of no sequences but if you if you were to know sequences complexity is not an algebra but this is somehow a proxy for projection onto that fact and sure the proof is closely analogous to the proof of the projected number and spies wages keep decreasing black the Sampras


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