3/6 Nilsequences


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3/6 Nilsequences
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Green, Ben
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Institut des Hautes Études Scientifiques (IHÉS)
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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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the I and the ones that get him on the room to room for you we don't a plan that may be what I would do is just festival recall some of what I said in the 1st elections so we talked about no sequences and Gowers norms and at the end of last time I stated the investor and the garrison answer let me just state that again suppose that as a function from the 1st advantages to see binded everywhere by 1 and suppose that it's Gallas its case goes nor is the Delta and then the conclusion is that ass correlates with nail sequence and there is a no sequence contraband which is an awesome watercourse function 5 of a polynomial sequence such that the average of effort and times carbon is Italy's stealth primed and here to well Delta primed is bigger than Serra and various complacency quantities connected with opined it so there is a basis for the finale algebra Paul G. Allen such that this nation of complexity that defined the default binders and the smoothness of flies binders I'm an iTunes 1 thing I forgot last time boasted dimension of Gee's Bend I'm so the way to think about this is to not worry too much about the last 3 lines service all of this and it's important to do things rigorously but should just think binded complexity as I said last time that this is a very difficult current approved but not the ,comma versus somewhat easier said today I'm going to talk a little bit about the contest which explains some hired a released a bits why no sequence is important said today I will talk about the conversation section and and the crucial facts here it is the crucial observation is that no sequences behave a little bit like polynomials so under favorable circumstances and I'll explain what that means and the derivative of the Note sequence is no sequence of degree 1 lover 1 so the derivative Delta sub H ,comma I have and which is kind and carbon plus H Bomb is in no sequence all steps of class best minus 1 if is no sequence of Class S I say in this respect no sequences while they behave like polynomial faces so for example this is a familiar facts for certain special cases and namely the polynomial phases so far I think Clive and to be easy to the two-part I Alpha and squared for example such acts and I mentioned before that this is no sequence so this is a no sequence all class too and in which the underlying Greek cheese just the rails I have lattice gamma institutes the automorphic function well at the polynomial appeared is our friend squares and then fly effects case the effects which let me remind you use each of the 2 pilots convex so it is no sequence of class to I was its derivatives in the sense that justify that Delta sub age height and his evil of powerful and squares -minus and plus square and the point is that the big stage let me just make it clear that this is a fixed date so the
derivatives is he had to the minus to call an H 8 times the constant Ch and that's a linear no sequence success a Class 1 no sequence for fixed age in fact this is just an attitude carries on the facts and additive parents so 1 a claim is true is that there is a complete generalization of this observation to arbitrary no sequences and many others mention another example this is the only this is a not very easy exercise some exercise and this is definitely a bit tricky convince yourself that so if Thailand is the old Alpha and times brackets Peter and then the same is true said Delta sub H and 5 and is the Class 1 no sequence and my mentioned this particular no sequence before they comes from the Eisenberg Group recall that high incomes from Heisenberg 1 and it was actually not quite Virginia no sequence because the the 1st flight the was discontinued the fire was discontinuous but this doesn't much math this isn't important set aside that's already quite a tricky exercise many questions so far OK does it explain why this is so I need to see a little bit more about these polynomial a polynomial sequences and so let me state formally a proposition somebody just described various equivalent forms of what it means to be a polynomial sequence so have BAE G but it's been a pretty filtration and suppose that the from that to is just some much and then the following or equivalent say festival is that she is a polynomial P is a polynomial sequence and in the sense that I find before so that is to say that he is a product and appear then is a product of GI onto the PIN where and GI is in ingenious of this career so that's a definition we saw before Annex is that the derivatives of P behave a bit like a polynomial said Delta sub H 1 Delta sub H K P takes values in G sub carried for integers K and all choices of H 1 and K and N In the interview I'm I want to mention so I I don't think I'm going to have time to prove place I mentioned this before says this is a quite nontrivial fact safer the BOJ satisfying 1 all obviously groups but the same is far less clear of objects satisfied so I'm not going to do anything gains of time come to give the truth windfall but I do want to mention I'm in there to other the equivalent properties warning at which I useful inside the proves I didn't realize what realize really on 1 other on and 3rd equivalent property which is 3 I that's for every and H the To the cake P of n plus a meager age as Omega ranges over 0 1 the so that takes values in something called the horse cross Cupid the values in a certain subgroups all from G 7 0
1 to carry called the host Kron Q group Page carrying subcommittee cheaper bullets anomalies tell you what it is so this is a group generated by but all elements of the former well 1 alright cheated the square maker is some so this is the element that defined by to the Omega sub song is equal to GE effects on this as contained in I'm again dominated by media and is equal to the identity otherwise and Angie lies in the system
element of this filtration so it is quite difficult to explain why this is important if I draw a picture make certain that more sense the future have to send rests see you can understand the whole squad Q greater dimensions 3 you can think of it as of 8 couples the group so 1 day I could do it so I could have G. G Cheney G G and Jeter so this is the element that I called G to the 1 1 1 so it's just everything on the every verses of the keepers G and here so for any tree in In Gezira such as energy whatsoever so all of those elements elements lost crop Cupid but you can also have things like this so I can have fun g primed cheap primed and then everything else is the identity so this would be achieved primed to the 1 0 0 and I'm like this the energy primed analogy to so this is a certain groups and somehow the combinatorics all pollen and sequences militant groups is intimately related to the commonest Horrocks of these keeps and had a multiply and commuted together understate its yeah take me a whole lecture to develop this but what I have ideas you have full printed notes on this by the way if anyone wants them I just send me an e-mail and I'll give you I'd have got notes for all 3 of elections that given this week sales there's a proposition about different equivalences of what it means to be a polynomial sequence but there is an ongoing too explain little bits what this so the statement delivered by having no sequins and conditions or favorable and then it's derivative is no sequence of Class 1 last so favorable turns out to mean that the also multifunction firehouse just a certain additional invariants so favorable means that fight is additionally invariant which we call having a vertical frequency so what this means is it's a client from the last element the loss nontrivial element of the filtration is a homomorphism onto the multiplicative complexes character continuous homomorphism benefit annihilate sir I wanted to normalize the lattice said annihilating the lattice intersect each of us that we say it's a vertical character and fine House has a vertical frequency at precisely yes fly all X's GSA is equal to CSI of GS 5 x 4 1 X in group and for rules central elements while for all elements GS in the G S maybe I should remark explicitly said and and G sub S is in the center gene and that's because I've got a filtration and so the commentator G with G S it's supposed to be contained within G S plus 1 that is by assumptions trivial ,comma because of good filtration of Class S so think of this as an automotive function fire that has some additional variants In the center and I gave 2 examples of the tall and those both do have a vertical frequency of both of the examples do you have the 1st frequency so you can call do a four-year expansion just in in the direction of this Central Group G S and to decompose an arbitary automorphic function fly into automorphic functions having a vertical frequency so by a have physical for expansion we can
rights fight is some overexcited all flights aware flight CSI size of vertical frequency and in fact while I I may come back to this images again latest safer for this to be useful we need to know some additional things about this so will need to know How the complexity of this find CSI depends on that of flying and would also need to have some rapid decay but these are the same all the ideas of the same as when he decomposes smooth function into its 40 inmates the troops said the same the fanatics just think that this is so just an additional invariants that it's not very difficult to obtained have in any given case any questions sold for carry-ons but so let me try and explain them away the derivative of and no sequence with a vertical character is no sequence of lower class so let's look at the derivative there is some there is not much difficulty in interpreting the derivative Delta sub page high and which is call I call higher plus a spot as the no sequence so in fact it's equal to 5 times fireball all of Van P event H so you can think about is no sequins on Gee Cross G so on GE crossed chief with filtration and which GE crossed G sub his GI Cross G. I and then what is it not is not completely trivial set indeed 4 figs grapes and P LE H is also opponent of sequence adapted to the same filtration on I'm cheap bullets because it is 1 is the derivative Delta sub H P the inverse times appear event and and his opponent of sequence adopted 2 G and so the derivative fashion by Part 2 and hence the product it is also a polynomial sequence it's a irreverent he's the proposition city center as it is in no sequence but unfortunately I have not proved anything like this observation here because the sequence is more complicated the situation is more complicated than that the 1 I started with simply taken a direct product of 2 groups of Class S so unfortunately I'm so unfortunately G Cross chief it is a bigger group but also with class so that's unfortunate but you can make some small modifications in such a way that this actually inside a smaller group said by making some small modifications we can ensure and we can pass to a smaller group so I claim in particular but in fact the sequence then P of endless H I can multiply that through by constant which is pretty harmless said appears Erie he evades inverse just check :colon corrects when asserted before I do that and minute the summit is 1st passed a smaller filtration therefore I passed a smaller groups so in fact this already the Pappy event Pimenta sage already lies inside a small filtration so I climbed so let's write the alright Pierre Pearman plus H is a event inverse time be abandoned where to Avon is going to be just the identity and then that relative peace and and event will be just Pierre and Pia so that's straightforward the I'm and both of
those sequences line much smaller groups so the advocate derivatives Delta H 1 HK event takes values in this writer all I had 6 values in GI diagonal which is just several pairs Gigi for achieving GI so that's obvious b is in the diagonal of G States derivatives will in the diagonal and the derivative called a while the truth is of any the ice derivative of a essential plus 1st derivative of P that takes values in In In the identity Cross she survived just 1 being and I must 1st derivatives of peace so by the product property therefore the products all that is to say the sequence here then appear endless H how this fight derivatives lying in what the group that you get by taking 2 the grid generated by these 2 groups which I will call GI I cross subtree plus 1 GI and that is precisely the Sandoval pairs Gigi primed with G G primed in and at least 2 things being equivalent modes GI plus 1 sentence ended that sequence of groups that I've defined gives you a filtration on the Prodigy crossed G and it's a fine filtration than merely the product filtration so this gives well it's not filtration yet it's actually a pretty filtration against a pre filtration hung on she crossed chief so that's an exercise it's really just a simple greed very excited you need to check that commutation of elements that does what you think it does put as the appropriate filtration property said to make a filtration I need to To get a filtration is a little tricky I which is most blighted by constant say we consider instead and the sequence of the film solids that consider the sequence he abandoned Piven plus H. This is what I started writing before times peers 0 of H inverse so too because it's just the previous sequence times a constant that derivatives are essentially the same but this also have derivatives so it also has always derivatives taking values in GI I cross some cheese 5 plus 1 GI a banana sequence itself takes values in In the 1st so the sequence itself will take values in that she won Cross subject to G 1 which is just the same various G cross-subsidy to G that's not difficult to prove let me just write it down right up this of half exercise Hoffa Hoffa natural proves it wasn't as quite as straightforward as I thought so approved so what 1 does is observed that some of the following P & plus H times P and inverse is equal to P H & R is equal to the derivatives and P of age inverse the times that derivatives and key of Syrian times appear then Pierce era invests so that's just an identity but the number this the Delta and pH inverse downturn Sara differs from up to a commentator reckons swap the order of the EC said differs from Delta and 0 Delta and PVH inverse by commuters which lies in that enlightened G 2 and and
this is in fact the derivative Delta H Delta and Peter of 0 it's a 2nd derivative which also lies entreaties so I believe a simpler mean that's not very hard but this Bromley a simpler way to see that on the Batman the reason I we have opponent of sequence which lies inside a filtration on In this group here maybe I should remind you what I'm trying to do and so remember what I'm trying to do is I understand why the derivative of the no sequence it is in no sequence of lower class and what I've done so far it intent the derivative no sequence as a no sequence on this group here now you can probably guess
that I'm not finished because I haven't actually use the fact that fire has a vertical character if you remember that I talked about this notion of vertical characterized not made any use of that year the chairman was there could I couldn't unfortunately still has cost S unfortunately G Cross G 2 G still has classed as and in fact the the S the best it's extremely hard to say th exposed behalf say for French people said it must be very hard to say yes 38 it still has processed in the air time In the filtration is um well it's S crossed as plus 1 alleges the diagonal of G S but it turns out that the vertical frequency of fight is precisely what lets as modified by that diagonal and hence passed to class best minus 1 group however fire has a vertical frequency CSI and golf come the finally Crossfire all of G S G S X Y fetus computer way is going to be the fire of G of S X 5 GMs why bond and that will be sigh of GS 5 x sidebar 5 white bother and of course that's just 5 times 5 are on the ex-wife so far I satisfy barred descends to a function 5 times 5 on descendants 2 function on on a group that I will call the square she said Jeanne square which is defined to be Cross 2 tree molded by the center might buy the diagonal of the center and this which has classed as minus 1 say is an exercise the EU if you if you want to check your understanding of these concepts when she is the Heisenberg group what the square exercise achieves the Heisenberg 1 1 1 ,comma toss all in the square is isomorphic to huge lies and agreed course is not isomorphic to work you OK what is more that needs to be said in this discussion but I don't think I'm going to say I'm going to the state of final what more needs to be said well 1 problem is that 7 so actually a slight inaccuracy here when I eat most played on the right by this constant had appeared nor P of age inverse I have to correspondingly modified the automorphic function fight Crossfire bar by 828 in fact to meet briefly sketched this owed invite needs instead consider and fight Crossfire part of X and Y times Penaud pH and US time summer yeah so
maybe I should have said that the new makes annotation words but the problem is that some no 1 said that these Pinault and pH of in any way nicely by and leave there no right variants of the Sobolev norms that use last time so what this means is that if P of Norton unpeeled nature very very big then actually the smoothness norms of this fight Crossfire could blow up and that's problematic he wouldn't be binding so you need do some extra tricks Is conjugating by elements of the lattice To make these elements more again but then things start becoming even harder to understand say I'm just going to say that the Rossum further things that you need to do and then stated final there since there are more elect high AT and T equals fly all of the event being and no sequence of Class S with a vertical frequency then its derivatives so the derivative Delta H high and which is kind and 5 endless H boss may be interpreted as that can be interpreted as a no sequence fire sub H box piece of H boxes and on G box which is defined to be the group so that's in that group of filtered group of class minus 1 it With the filtration mentioned before and son-in-law With more cash and with some additional arguments L 1 Maine shore 1 may find that the um the smoothness norms all 5 H box and in terms of those of fire nicely intoned terms of those falls flat so of course the more precise statement that needs to be made that have that's that's the basic idea and doesn't want any further elaboration of any of that I'm guessing what he well there's a lot more that can be said that if you if you really want see I have I have always notes on giving the precise precise spines for all of Bono and explain the now is why this fact senators saying that no sequences behave in many ways like polynomials why this implies that they are obstructions to what we call Gallas uniformity sir in other words if you correlate with no sequence then you have a large Gallas norms so that's the contrast to the investor there are for the Dallas norms it said in fact what is useful to introduce something called the Dallas jewel that we introduce the June norms and the June norms so the June on although functions beside UK and stomped days by definition the Supreme Court however With this was banned in detail Standard the site superior away with binded gathers no UK and equals 1 all the inner products of that with excitement so this is
actually this is just the standard construction of the jewel in all this is that this construction works for any normal practice and it gives another long so no 1 since then and I the comments to the investor together on this the statement that no sequences have a small Giorno that's the theory as soon as measures we have to apply is no sequence and the June long of Class S and the dual s plus 1 is bounded by a suitable smoothness norm all 5 5 on I forget I would that at some point ,comma you need the number to read that you need depends on them it's not terribly important but the the number during is unique does depend on the dimension of the Group G and also honest I think TTS times the dimension of cheesecake to be spin-off so this statement is true well you also need to take account of the complexity of the the Dumont ,comma so this will depend on on and the complexity of the Nancy but the basic idea is that afflicts no sequence has a binding Gallas dual norms and therefore that and no sequences obstructs Gallas uniformity for the Spanish side even more correlation with the no sequence implies launch cows north and the way the numerology works is that if the no sequence has Class S and then the Gallas Norman is the ass plus 1 so sometimes it's difficult to remember the numerology just to remind you when I was interested in 410 progression arithmetical aggressions I needed to talk about the gathers 3 normal use Renault and to understand the Dallas Usrey norm I care about no sequences of class too which adjusts the step up from a billion and to work with 310 management rations it's the is 2nd on it's important and that is related to know sequences of Class 1 which is just a billion for analysis therefore the truth and 1 1st gained himself a different formula for the Dallas norms and she this is quite a suggested format I think this is what tells you that the gallows normal has something to do with derivatives said 1st of all and said that the gas norms he Gallas K on and is equal to To the power to to the Kerry is equal to the average H 1 H K and acts all the derivative
where dozens of age and relax it is acrobatics Aphrodite frustrated by some as nothing but a restatement of the definition but it makes it sort of a bit clear derivatives have have some role to play and hands you can use this to what inductively with the cows norms said the Gallas use K plus 1 or to the power to to the Cape plus 1 is the same thing as the average at the age of the UK norm of the relative to the power to care so that's a useful facts since then this fact implies the corresponding facts about the dual norms and it's that facts that we will use to prove this theory and that's really very natural because we understand that derivatives and then we can prove that served by induction on so on the dual side I claim that I might claim is that the Gallas June normal UK plus 1 "quotation mark stalled it's kind advice the maximum the Supreme over each fall of you'll norms of the derivatives To about one-half and to work with me all OK so what's ahead we prevent I will let the arbitrary and then the inner products evolve backed with CSI it is the same thing as you gets by taking the inner product the input of rough with sigh Square is the same as taking the average now I should say that there are there a couple of technical issues here that I'm overlooking what I write is correct if I replaced this 1 2 and by the group said and and then all averages around the whole group I'm not taking averages of the whole group I have to be a tiny bit careful but let me ignore said this summer this is just inequality just expanded and then this is going to be blinded by the average of age also the jewel Norman all of society sometimes the the Gallas norms of that just by definition the jewel and this will be bounded by the supreme over each of the jewel norm times the average
rate but this guy was known Of the derivatives about and then I can take hold as inequality let me raise that to the power to to the cake so that's the most supremum over age of the jewel Norma that derivatives times the average all this to the the case to the 1 overseas to the care and that's holders inequality and that is basically the end so this equals stopover H times the Dallas UK plus 1 Over set out said a proof of the statement that race at the top Let me see if I can just get them back that's what something this is of stock so basically already told you how that proves the theory so what through the serum by induction modest steps the proof but before you go and apply the induction you 1st need to expand finds a vertical characters say expand Pfizer some or all of vesicle of automorphic functions without a vertical character Haske size has a vertical character and then so they just by the norm property the jewel norms of this is that most of the summer the side of the dual norms values and then implant then apply stock
but sorry I didn't mean was known to fly I mean the guy was norms of fire so here acquired concept sigh and demand is equal to fly so excited of the event the now play stock the by induction and induction on so by induction on the class S and the facts no sir Correa sub side has closest minus 1 I would dump so that's that's what there is to it and vertical for expansion into water moving functions with this additional verse :colon variants by the center many observation that the derivative of the Note sequence of Class S is of class minus 1 and then this fact so he is an exercise so if you didn't follow any of what I said about groups infiltrations and so without any of that you can convince yourself I that the during all of Eastern to fly Alpha and squares in the East Regional disbanded and the thing is that that's already a vertical frequency so you don't need to do a for expansion and so we you need to do that a supply start and then induction but said it's almost 12 o'clock 1 I'm going to do is just tell you I don't have any ones still could be here next week that good the on maybe I can help you choose which dates ,comma so try and make as inasmuch as I can 3 lectures that are independent of 1 another and relatively independent of what I said so far there next week so much for I'm going to talk about the distribution of properties properties some of sequences so this has something in it somewhere halfway between kind of traditional uniform distribution problems in analytic number theory and questions in ergodic theory so the basic question asking is is P event for adolescent and close to equidistant beaches In terminology so this is really a question about the good distribution on a homogeneous space they say it's kind of fun uniform distribution theory theory meets a ergodic theory and then in the 5th lecture while goods and talk about the 2 applications that I mentioned starts in the 1st leg draw talk a little bit about similarities there and related issues and then in the 6th lecture I will talk about it some aspects of linear equations in and that is where we use said this to talk about distribution properties 1 does need to use facts about pollen sequences again the largest I will recall me what I needed and then to talk about either of these 2 issues 1 needs to use the guy was norms ,comma and also things from here and from what I've said this week other than that I think I Oh stop that but let me just remind you but I have here when debt uncorrected at the moment but some as always it's a private thing if you just e-mail me I'll send you my uncorrected notes so this means that if you want to read over the weekend Everything I've said with all the details from the 1st 3 lectures then you will be able to do so said sending e-mail and my e-mail is banned adult green acts and enough told OCS thought took very important to know that it's better not Benjamin because there is somebody called Benjamin Green adults would and that's a different me he does not have these notes that became and services for this week is this is here introduce were there was a new user of the reading this the waves question the use it is useful lists that's an interesting question um it's quite unclear "quotation mark clears me whether that's true or not I and I I would think not and in any case I don't see why would be easier In free option it's not even clear to me the sense in which an arbitrary filtration can be seen as equations coming from the freeing of 2 degrees because infiltrations can be a lot faster than just below a central server that can be very for the week of operation that the you have some sequences see season in U.S. no well all of those things are true and in fact In that sense you don't need the nation of opponent of sequence or arbitary filtration at all because under a polynomial no sequence can in fact be written as just a linear no sequence 5 of GTE adapted to and this is automatically adapted to the lower central series filtration on some grids so why did I not do that what the point is that when you come to think about distributional properties you can't guarantee that this is nicer distributed and if you it's very important to have a theory in which you can get a nicely distributed no sequence and to allow that I have to work in this big a category of pollen sequences so it it's not enough to interpret your nose sequences living somewhere else you need to initiate war with information about the distribution to move to the new setting as well and it generally I wouldn't this To 230 on Monday W


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