4/6 Nilsequences

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4/6 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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it's been
a year of a new Thacker Her there around the Internet on and on and on and on and on and on and on but cases Grigsby talking about the distribution of equities to all and remind you just quickly what no sequences shortly but really needs be talking about generalizations and very very well may in fact the generalizations and quantitative very old the following very well known facts which is that if the real number then the sequence about these things said to the history the exception it is equally should be modern ones this is rational the generalizing that 2 polynomial sequences on arbitrary I have questions Dumont ,comma hero just got onto and I'm also going to be interested not in the whole sequence up to infinity but in finite segments of it and all of that with a mind to applications next time off so let me just remind you again what this thing called no sequences called we have since no which has only a proper filtration the bullets collection as a collection of closed lease subgroups of G which satisfy this conversation inclusion proxy that I mentioned before and then I define this notion of a polynomial sequence we find in several different ways at which demonstrated the equivalent but I stated the nation of polynomial sequence here on PC from said so I won't go back again to last time and you don't really need to know what the general definition is that the basic question is is when it is that it's a colossus gap let so if I have a finite set of values of that Poland when is the event from an equal 1 big and close to equidistantly In the quotient ,comma much of what is this so it's a question be interested what made this medal give you a few examples already have 1 let me festival said that the example I put up there actually generalizes this is perhaps the last 1 in fact generalizes to all the proponent of sequences he called on and get on with it and it appears just any polynomial so I think I I briefly mentioned last time has made the traditional notion of a polynomial compatible with this slight consternation to do with filtration if the polynomial degree d he just take the 1st the elements of the filtration to equal the whole group which is on this case the period .period than their peers it the sequence in the distribution and if and only if the parliament has at least 1 in rational coefficient other than the constant the court it is not safe travel to prove and I said the words about it later approved uses involves inequality which I will mention later there was inequality would tell you that if the parliament was not secret distributed then the least efficient is rational and then you have to it's right down to show that all of the other coefficients are a rational as well a season of inequality and alliteration I am citizen according the move Leon agreed the relation of my motion .period this is about equal distribution of flows of militant groups said she is no patent I simply connected with group and it appears a linear sequence the man is a qualifies as a polynomial sequence with the definition of given then conclusion here an equal quantity is equal distribution what is a good distributed meanwhile B carrier that surely but let's just say for the moment with respect to the hommage With respect to ,comma on government but while in years the anyway that sequence can fail to be good at recipes is if it fails to be good distributed in an opinion projection of cheese
the appeal projection of fails to be distributed so what I mean by that is that you have and you have a natural projection to GE mode it's coming the sequence he projects that so this will be isomorphic to adjust the good talk when he projected peak it will be isomorphic to assert rotation projects to much upon compares with P this induces amount on Taurus beginning example with highs in the group it seems that there is it is equal to if any of the previous projection is secretive should be accepted I think that I for example G's Heisenberg with you I made just some 1 1 1 Alpha Beta Gamma and then the flow induced on the two-dimensional tourists but that is just rotation by offered B. the Inge's not on tour Tom said square is given by Peter and in alpha and beta so you have no means there is the nontrivial 5 the sequence In the note group the secret distributed in Romania this sequences he could attributed the Taurus which by chronicles the armistice I'm as saying that Alfred Peace or independent of the rational Alpha Beta and 1 independent of the rest of by chronic this is the question if 1 of these are linearly independent Q so that's a complete criterion for when paying no rotation on the Heisenberg groups speaker distributed I'm really going to be interested in quantities of variants of this and even to say what I mean I should give a bit more discussions 3 questions so far quantitative coordinating quantitative you have to quantify the notion of claims that may introduce a parameter I don't agree with what has been happening well if somebody asked you how should I define equidistant featured on the circle probably the 1st thing I would say it is it's a sequence that spends the right portion of time management but that's slightly problematic to use that as a definition because what should be the of interval in a more general group as where can acquire like that but we think of millions of older characteristic function of the interval a function as it is and then he could distribution is saying that the sequence of functions are close to the end of that function we say a sequence he will always be thrust upon a sequence taking values of the sequence speed is dealt with issue the government so when you look at the average of the sequence weighted by any functions on despite the functional gene would gather and then you compare that to the integral with respect to Hamas that's the most Delta times inappropriate normal function which North America Way here every day for every new automorphic function on we put a on government G injuries from the new all on and then normalized in such a way that the quality of this question 1 so the match the whole space 1 and the time it is not too difficult to understand the Heisenberg group it's actually just the XT widens and Eisenberg integral is simply with respect to the 61 these are measures just the product of all measures in this report so you would lose much by thinking about that case of the Circle K and and the which only I mean if you change the moment will change the notion that you dealing with but in relatively minor wife so the technical convenience it's actually useful to have the honor to be 1 of the top norms that I mentioned last so what so you just take it to be a sufficiently high sublet Norman I think I found that
the subliminal with 2 CDs times at the mention of G derivatives with that's really technical must invite you could just look at 1st derivatives of fire and every function with find 1st derivatives is basically shit when is and it can easily be approximated by much smoother functioning so is really just a technical point and the basic idea here is simply that you are comparing space averages and time averages and that's how you decide whether sequences place we could questions on so crucial results on what has been the main result but I want talk about it a more complete statement about when Napolitano's sequence on Dumont ,comma if they could distributed and it generalizes is really the examples that I showed you above a generalization of the examples 1st of all I had to put sequences on commentary and with the quantitative notion of distribution no 1 should sir Leon greens there is is valid for any simply connected militant group and in in linear sequence on group if you don't want the polynomial sequence as a more general result but it turns out that actually polymers sequence can be lifted to a linear sequence on a much bigger groups sir actually get polynomial he quit distribution result for free from the linear 1 but for various reasons that's not bad argument fails you in the quantitative so that I mentioned but I don't place any emphasis on so what you want to state precisely this may result in the sketch some ideas of the here is reasonably priced but it seems to be a bit of a theme in the elections that the most I had to do with precision state results I guess sometimes difficult said they should be as above let's suppose that appeal fails to be dealt equidistant from 1 to 1 what the experts an abelian explanation and all right and what it is there is a horizontal Paris so I'll introduce a future that I will explain later horizontal character from G to the real so what we call the smoothness and all the Europeanization of pain His binders banded by delegates to the demand constant whether constant depends on so I have a lot of things here to explain but if you don't wish to take too much notice of what my explanations this is an obstruction the sequence being John Paul II is projections from G 2 the 3rd added with with the well defined here the the pie this will automatically be so I just have to tell what horrors of characters here horizontal character just means it's all office so automatically annihilated the commentators subgroups g brackets Street and it was supposed to annihilate them and this move in on the last composed with pain well it's going to be an almost said valued polynomial so a polynomial not pulling back from the integers two-on-one said and so just write it in the usual way right and and this was the form after that is often or place of 1 plus out of the hands of the and the smooth this norm is basically telling you how close to being rational all of those confessions statement Back at its smoothness on the Infiniti His most needed what it means that these conditions are very close to integers means that after Jack In all modes which is the distance from to an integer is most on the job for every Jack so that's a lot to here let me tell you it these hot specializes .period too just think it's a bit sad but before I do that I said that this smoothness norm is telling you that these are forgeries close to being rational and she's telling you that close to being integers 2 that's enough because you can always multiplied each through by appropriate integer and make things at the close the rational close to integers let me give an example example just which equals are Democrats the sequence these square in an equal 1 sales to be down to equidistant if you have the very very close to rational if Romania there is key In the interview is such that distance from key features to the nearest integer bye the authors of the money
constant and square where's the cue coming from will multiplied modification by did here is the same thing as 1 of these horizontal characters right now the east it multiplication by this is a result of this follows from above inequality but this is not the usual way of phrasing often ideas the way I prefer to monitor election on balls and quality of 1st stated this way find the usual 1 little bit confusing didn't have anyone familiar with the statement of the inequalities that is a statement about certain exponential sums with :colon phases but it's really saying precisely this Parliament of sequence and when it's the question that is something lacking in but a source final here as stated has no contact whatsoever because every time by despite the suspension pigeonhole principle for anything he could find a Q him I have to be very big such that the sides here is close to an integer so additionally need a quantification and to keep this country and we also need to place a value on the complexity of this horizontal character that certain complexity "quotation mark complexity of and that was before this has got to be measured with respect to a basis on the layout of the that is bounded I don't want to go over again what the basis was on how you define exactly this notion of complexity but in this particular example adjusted generator the size of cheese for example just give up it's as if the band on his also he should not be otherwise it's again completely trivial sets a ban on the sides of him in this case while the size of year's again bounded by Delta to some power that is the 3rd precisely I would need to include also and some banged on the complexity of this horizontal the Limoges rice what the content of this service in 1 phrase and he felt that could fall in the part of the way of the people places idea here let me just OK that's a good point the notion of equivalents in the sort of discrete setting it is always a bit you don't quite get back to where you started from said there is and if and only if but if you gave that way and then back Delta would become Delta's system backup power so that phrases it is a basic idea eponymous sequence the way I have to be good distributed in her opinion .period a little bit about free obviously not very much because this is really difficult to prove but the proof uses a trip was more than triple a technique that is the same technique that uses the profiles quality and that score Vanderpool puts less than main idea Peru the following vile and this is not so follows voluntary the paper and analytic number theory is not the same thing that Green does in his place and by following where will we use was "quotation mark founder called them so the hope and the relation between sequence being condition what the original band called the original version of 1 as like between them and and the distribution of sequence the uniform distribution of sequence you want 1 and its derivatives which are you you specifically for of derivatives so we could distributed and so the sequence that would actually immediately give you qualitative equidistantly should results about the threats square because the evidence will then be linear sequence and then you can just apply the results founded in sequence here is an equal and corporate inequality which is very I suppose you have just bandits sequence suppose that antique with 1 and on complex numbers normal 1 I suppose that there averages not place is supposed expected value of the land is still that conclusion is that many of the derivatives have the same property some of its stated precise questions the onset of an approved staff officer the suppose that big a slight technical condition begin flights between big is neither incredibly tiny nor quite as big as then we have that the average value all the as the stone square for for many values about the 4 the 32 month for Italy's don't
square with the 2 kinds sometimes I find it less confusing than if you agree to put an explicit constants rather than on specified absolute Constance when the not too the statement is if you've got some bystanders sequence then also many of its derivatives the sketch that I would think small and that going case be guaranteed by the inequality that averaging it is very close to the average man shifted by it I mean they're averaging essentially the same things that you got some slight edges so many of those averages are while is based in the Kurdish quality so you can choose just phases eschews and unit complex numbers that are each expected values and new 1 of the handful of stage he the Delta to let's say for an appropriate place and that's just to make it real if you have responded average have reached the age of the average unless 1 all of them I have haven't is also list and now you plication by which the average of the average age of the word eyes at least don't square foot and if you expand that that implies that the average H 1 and H to enlist the of the average of analysts refer to 1 of and play and H 1 and states to bar is least sweater for when you can see that with a bit of messing around you get the statement that I wanted every average on the inside here is basically like the average in the statement of Vanderpool this inequality this is roughly the expected value of an analyst with an eye and a and Class H 1 to bomb and then you just have to count how many times each H 1 1 6 2 except but that's the basic idea is just 1 application of culture shots after done shift question the other he is also key idea is due to fog which is to understand the contributions could you should expand into forests the crucial idea of idea of proving he quit distribution by expanding or put another way we had a definition of the Quidditch abused which side of the road but the definition was about an arbitary function fine and you need to check that definition on a basis of what else to get check the definition J. definition of equal distribution only for Fire Line basis all too Of course is not quite enough to just check things if you're dealing with a quantitative questioned you can't just check for basis and then hope that the same statement will hold when you add raisins almonds together but as long as your way decomposing into these basis elements is suitably effective which it will be at the functions are highly smooth then that's more or less the case what about famously In the case of all modes it's enough to check enough to fly balls to be in the 2 III and an Indian and as said you do lose a little bit in the delta this causing some slight degradation and slight change in just a power but that in the case of no sequences in just 2 2nd but let me explain how it works already in the case of safe the squared so if you want to test the the dish distribution of the and squares on the circle is enough to test it for each of the 2 parties the current square lavish happen to be larger than you could apply the founder Copa inequality and get that many derivatives that have have large average but a derivative of each of the 2 parties and where it is just linear phase function which is just a geometric series so it's quite easy to evaluate them so what's appropriate
variant of that this for expansion In government G Jamal down wedgies and no group will cost that's a very deep question in general but actually it Delaware Valley simple answers that when going to look at a sophisticated way of decomposing the space is that she just enough to do for expansion in the vertical component the general similar works how to test equidistant region for knowledge refined and decomposed into automorphic functions without a vertical frequency equal to against an arbitrary the Infiniti ,comma launching to handle the case when fire vertical frequency CSI which reminds me that I if he too expected by an element of the law group in the filtration plant has a natural transformation properties the reason that simply is that there is you can do for expansion in the direction of the Group G as I mentioned last time fight and expanded all automorphic functions with vertical frequency and of course all of this would have to be done quantitatively by John boy you with the details of since you don't suppose that you have a function that additional has a vertical frequency sighs and non-trivial frequency and that means just as with the nontrivial exponential from circled the integral this function with respect to the harmony because of the farcical oscillation adjust Council's acting you so what is the Quidditch wouldn't think equidistantly distributing me in this case the appeared and from any cause once and fails to be dealt to equidistantly With respect this particular automorphic function with a vertical frequency then it just means that the average is down well actually felt at times the smoothest normal function better the prospect so this is something that we can apply the founder corporate limits my call such but now with an additional 5 Piven at least some calls sometimes and felt the square now I don't have anybody remembers what happened last week but you can see that something very similar to what happened in the truth involves inequalities have evolved quality From the corporate took a quadratic face functions and by differentiating it made and what is the derivative of no sequence why spend all our explaining that that is no sequence of Class 1 left so we can turn this into an induction by this device interrupts by the main discussion by the discussion of the EU's mainly election 3 the object here here and fight appear ball 4 figs dates for the state no sequence of class minus 1 means you can imply you can hope to prove the name there by induction on class this offers the possibility of proceeding by induction on so this is very special cases the Snowflake 1 can consider flows on Dumont ,comma in other cases for example when Jesus still to the Gamez of subgroups but the same sort of trip just wouldn't work if you get this would not be a simpler objected because it would be worse objects sitting on G crushed but in no case would you have simplification so that's the main idea of approved but you still have to actually carry the induction of carrying this induction rather difficult
and to sponsorship just give you a glimpse of what happens in the Heisenberg case so you can see the way in which it's difficult on the manager of complete the notation of last time that kind to be no sequence and then in the administration had lost money Delta ,comma is precisely said that this derivative is known to the process minus 1 but while major let me specializes the case of the just the illustrations g is high and that the polynomial just a facade and linear I said last time last time week but the derivative Delta each high all of them can be thought of I and no sequence of Class 1 and I retire precisely what they the Director General for piece of a Delta In the event that each of those people and payment of the times .period peer baked in and that's actually In the linear case is just the same it's just the diagonal flood aid to the end of the year and the linear you can't blame for easy calculation and the automatic functions fine flights books and was flying times fireball of X Pinault PA which in my case is 5 x times 5 timed to the and here we come across a serious issue which I alluded to last I'm trying to prove a statement about the good distribution by induction but the standard enjoys prove quantitative it cares about how smooth my functions file Lipschitz but unfortunately this is the age could be huge almost nothing about a spy this could be an absolutely enormous government Group G. so there is no guarantee that this autumn automorphic function has any nite control too could be huge the age could be huge and control on 5 square so the induction won't work unless you do something get around that issue we need to do is to insert a conjugation by elements of the last ,comma together right we can come by elements all across Canada to many artists show has modified Institute and you could put a Yukon conjugate despite all the trees down damn and this has the effect of adjusting the automorphic function here so put some money to explain this I didn't computation last time explaining why the derivative of no sequence could be written this fall it turned out that there's extra flexibility inside evidence and make use of you can do this conjugation in any way you like and you have a different representation of derivatives in the form and this will then become Delta age I can take ,comma North equals criminals the identity so I get to multiply ACH by any element of the losses that I like on the left and by doing I can make it by choosing she's gonna that I can answer that lies in the fundamentals so particular while its major countries battered by 1 and so when you do that you do then recover control on the social function unfortunately you made the problem more complicated so you have to do this to make get yourself a derivative function that Scott bandits smoothness norms but you may be polynomial the derivative of the pollen and more complicated Peace and while its rotation by by the diagonal and conjugated doses of age comes of age I got think in the so 19 implanted inductive hypothesis that is just a rotation on the toss the rotation of I I introduced this last time that's what you get by applying the construction schedule after husband it's a three-dimensional Euclidean by car
we have well essentially the coordinates of the event the controversy fail to be independent of the key the coordinates of government we have to while reduced to life the story reduced more they're being the key for many families of Susan billion problem but it's a rather complicated 1 so you have to somehow go from the assertion saying something about the coordinates of the self difficult matter you go from there the statement about it but it can be done and that it was always going to be more details the paper on which this is done it's difficult but the fuel ratio that you kind of know has to work if that where for about distribution of sequences is correct and that's because found corporate almost indefinitely it's not you haven't really lost anything apart from a few pounds of Delta and going through all of this machinery the expansion into series against more than so this is lower way sort of integration problems you taking derivatives and you have to deduce it was a victory but the statement you ended up with when you did that is quite difficult to integrate back again and I should say this it's the same problem with the proof of the inverse conjectures for the cows and I showed you last time y White Mills who considers or obstructions to guarantee uniformity by induction size said if a function correlates with no sequence kind many of its derivatives correlate with derivatives and then I was done by indulgence but to get ,comma statement is much harder so many the derivatives that correlate with no sequence but you have to then deduced from that act itself correlates with announcing a similar sort of issue it's difficult you it is all over the place such yet have 4 In the case of the you for Normand this is used everywhere I didn't use the inverse conjecture .period here this is purely an almost an algebraic fact said nothing about norms you know for the full Norman you will be for it's needed this is routine this is somehow key to the entire theory I for analysis and no sequences is also needed crucially ensuring that but tomorrow I'll be talking about crime and you need to show that the matter function is orthogonal regional sequences and again this is a crucial ingredient some quite fundamental what is the shortest 2 things about how this is applied way I recently about why should care about equidistant additional sequences Hawaii to attend about this particular motion picture distribution of polynomial sequences on government and the first one is that essentially every sequence is a Christian every is almost equidistant I'll say a little more about what that means in amendment but it means that peace and all that tree pollen and sequence can be decomposed into some very benign pieces of periodic peace in a very smooth peace possibly could distributed pace the of distributed cases pretty general and secondly it appears that the secret attributed the weaker and weaker various configurations so if I have time I'll give you an example to the average falls I think it's a four-time progressions involving no sequence plus 2 I told him and 3 it will be it I very algebraic answer is an integral for Harris subgroup called the whole
Petrosky Group H P 4 chief Martin this way take the floor of the of subgroup Christian Christian this group explicit grew cool the whole hold just code for and actually that this generalizes is nothing special about athletic aggressions there's corresponding result for any linear forms which is "quotation mark lead agreed I wouldn't tell you or hold just agreed this because we've already seen action in June when she is went all but with the filtration so it's but a degree class to filtration on G and then I just go for HP for GE it is subgroup of all 4 and it's the set of toppled X naught at 1 6 2 3 such that no 1 3 6 1 plus 3 6 2 plus 6 3 is equal to 0 we've seen before
and because I order when
Parliament sequence would just be feature Tehran squares and the what the example shady before always when was read to and give an example and we notice before that there's a constraint amongst Fortum progression In this function I mentioned that when I assure you traditional for analysis is not enough to handle progression and what this term is saying is that essentially that's the only construct so saying that the Fortum progressions all inside the dysfunction equidistantly the central portable satisfying the constraint In a sense and squared routes to London and to reimburse the square root to class 3 and close to the square root to command and plus 3 square roots here because it is the end constraint on water aggression for in here so he could distributed sequences of I feel like slightly falls you actually need a slightly more elaborate nations stronger notion of a attributed pulled the rational it's quite technical and I I didn't feel inclined to what it is In many cases for example for the the linear sequence is on the horizon but agreed the 2 nations coincide I'm going to explain what I mean by this statement what I mean by parliament sequences almost equal distribution I want this is basically a generalization of what's called the hardy Littlewood math said Sahadi in would were concerned with what things like polynomial faces on the circle when they're thinking about wearing problems and they decompose those faces for the coefficient of its faces into major and minor rocks and In other words they basically said every every numbers I essentially rational not rational their action and that's essentially what generalized by the session so let me explain .period 1 it's really a generalization all the heart what I of decomposing in the major and minor off and in fact about this Hollywood wouldn't it can be seen as a special case wait years the circle accuse all of this so more precise statement all parliament speaker I can write it as a part of cycle ride he added that sometimes people at times this is point was proper of polynomial sequence where and sometimes very very slow new so that means something like please comments above all the group measured on the group in some way out it's just a very very much so this means essentially that you can always some standard tricks analytic number theory like posh information just ignore it and then again is Russian and that means basically ,comma modeling lasted for some small key and then again in practice I mean you can get rid of gum despite all despite splitting into some progressions of common difference keys and then finally appeared after he could distributed highly could distributed someone but not necessarily G amid the peak could
just trivial pickup be identically equal 2 could be the identity always and then officer you're going to have a hard time send anything related to it as you could attribute Jr but people that is highly could attributed someone on this summer energy prime what can and primed contained in Chile a close connection subgroup and the statements a proper quantification of this is quite technical but you need to say what you want to buy highly quota should be you get to choose it to an extent that at a certain costs and then you get to control the complexity of the G primed eccentrics so there is a much more precise statement that needs to be made but this is the this was transferred there in the example I mention where P is constant this decomposition is true to say that someone is also constant Gamez constantly the primed calls and it's highly could distributed when she practiced identically what the identity and then the other extreme would be what he's already could distributed and then you could take a Solomon ,comma to be true it's just let me explain it has proved at this point is why this is the reason that we have to deal with polynomial sequences rather than merely Union what the witnesses presented by induction on with induction on the dimensions so if he's already could distribute then you've got nothing to do stop otherwise the main there might be talking about today tells you that there's an appealing explanation of facts and what this means is that he basically a sits inside a proper support so you can get to induct downwards on dimension that will eventually stop the is by induction dimension using the distribution of what can happen when you do G the game with the horizon suppose you take appeared to be a city a linear sequence where is the 1 1 1 3 2 1 0 and hit the is of size about 10 to the minus 3 halls so you can compute what end the end is 1 thing and 1 1 and and the whole times times and minus 1 now I played the air and equals 1 2 is not closely we could and we know the theory that there must be an abelian explanation but in this case started in the face the the extraordinary never really leaves it's visibly not it's not close could distributed because the Juventus stock near variances the survival find this idea that was induction on the dimension let's restrict the whole setting to the group where taxes but then 8 and is no longer a linear sequence it's than a polynomial sequence With the century so this can't there's sequence appeared into a policy sequence upon restriction to sequence so that is the polynomial sequence on an abelian Torres so what is .period shows is that so you have to work in this category a polynomial sequences in order to be able to make any of this was just the latest sequences that have enough closure of this decision the group property a polynomial sequences again when a and Times Beach and is not a performance sees end In another billion world a case in point is the 2 lectures left undone they will only tangentially rely on this and not on the details of this the next time I'll talk about somebody's there cases of arson some variants of that and then lost lecturer will talk about applications to the prime but question 2 so you but 2 1 2 yes and you to do that you need to variable version of the contribution instead of just the sequence index by any sequence in by and and I actually said the paper which these results will prevent page concentrated behavior polynomial on no manifolds annals of mass 2000 what's 2012 but this paper was written in 2006 and recently we discovered that this was wrong the proof of utmost importance of options it's not corrected but it was wrong for 6 years actually enhance actually were innocence the proof of the inverse conjectures with and it was wrong but flexible body yeah that's what


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