2/6 Nilsequences

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2/6 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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name my my my my had him Oh hey my eye a and we walk In the end of the year you can also I'll stop by his recordings 1 or 2 points from last time so last summer introduced the nation's of gallows norms these are norms on functions I'm going to go over exactly what the definition of those again later and I was too talk about no sequences which are the main topic of these lectures in fact has let me just remind you what I said about what no sequences said no sequence in fires he abandoned plans and the From center jail is a polynomial sequence and she is some s step whole class what is of what's the French term for class all use class class no patient group no presently group which ideas with I'm going to see this simply connected and fight is automorphic a smooth water more fake function gambled move it function for some molasses and gallantry I know I spent a lot of time in the 1st lecture talking about aromatic progression and I I mentioned stated something called generalize from 1 of their which provides a link between accounting errors most progressions and the guy was and I use that to motivate the following questions so when it is a garrison on plan is the Yukon or small because if you can show that for a particular function enormous small than that function doesn't contribute to counting arithmetic progression so what I want to explain in detail today is released the statement of the answers to that question traveling led the state rough for us sits there and this is the inverse of their for the Council they say elects suppose that sends space doses greater than 0 and suppose at his abounded functions from 1 of 2 ends complexes I suppose that it scours Norman Is it the stealth and suppose Back in UK about his early stuff then the conclusion is that correlates with no sequence and then there's no secrets archive and the equals 5 and such that the average value of effort and time Scriven Israelis don't surprise me where he Delta Pride is bided away from 0 in a way that depends on Delta and can set again normal function is large then it correlates with the no sequence now as I've stated here this result has precisely no content that's why it turns out and the reason for that is that just every function From 1 to end can be written in this for kindness 5 people In fact with just taking cheek was off so only that is as an exercise and so do add content you need to say some additional things about kind so here is it's in fact a minus 1 a class came on 1 mill sequence and then its complexity With complexity bandits In terms of just delta k only Albright said an even more trivial reason why the 1st statement have no content is that I actually the way I stated it to begin with a wasn't assessing and banged on pots so I could just have multiplied K bytes some huge numbers this would be true to among other things complexity controls the size of cash and his accomplice to this conversely well if S has a large in a product with a garret where there are no sequence then it's Gallas norm is large ,comma if call is a class came on this 1 will sequence Our complexity well you know said that city's if Kaiser came on this 1 but lost him as 1 no sequence that correlates with then has a large errors so that in the guy UK norm is Italy's Delta primed when I dealt a prime depends on delta k and all the complexity of depends on Delta carriers and complexity Of course so what
Monday aims in this sector is going to just be too at least tell you what complexities and others and I don't think this is ever been done in a lecture by anybody and so released but we will see a rigorous statement of what the investor for the cows norms the point of this is that somehow these no sequences off a complete list of characters if you like of the sequences abided complexity of complete list of characters for this high of Fourier analysis which is supposed to be city with these Gallas norms which themselves control the aromatic progressions and all the other related questions and let me say a few words about the proof of this so this is actually very difficult and long and it was finally finished off In a 140 page paper by Thailand's even myself and 7 special cases were done earlier unearthing the consensus is that we don't have we don't really understand why this Istria we don't really understand why no sequences or the complete list of obstructions to the garrison small the Congress is much easier and I'm going to tell you a model it probably even sketch proof of this so it's relatively easy to show that if that correlates with no sequence than the guy known as large that so many questions before I set off on trying make all of this more precise good yet so the plan is I'm going to just find all of these terms properly so let's define the gallows norms festival however it nicely what is best defined the Gallas norms on on abelian group festival rather than only into 1 2 and selects letter said be an B group finite abelian group and let K greater wanted to an integer and then we defined but the gallows norms while Britain's 1st of all defined too to the case power ad in use the carrier Z to the 2 to the case is defined to be this average so the average the X and then over at H 1 and H K all the products again Omega ranges over the end user 1 to the end all explain this moment Kelly the Omega act of X plus Tamika . 8 I say here Kelly seized complex conjugation it is a complex conjugation and and modern again is just the song 1 plus 3 workers perhaps to decay so let me remark as I did last time is not obvious that the quantity of the rights is Real and negative but not actually turns out to be the case so it turns out that the right side right outside by the way you should I feel very embarrassed that my French is so bad that I couldn't even dream of lecturing in French but if I say something in English that makes no sense like yesterday use the phrase red herring which apparently makes no sense whatsoever in French so if I do that again please just tell me again as the right hand side is real done non-negative I'll it does not quite obvious is not particularly difficult either but it's not obvious and does so there's a unique choice all 2 suitcase reach which makes that that norm Real and all negative a defining alone ash in the UK have said to be the unique real and non-negative took case Ridge said depending on how much time I have in this section that and I do want to get more substantial things I may give some indications of the priest of these facts which rely on something called the Garrett Koshy Shrontz inequality the main school in all of this part of various justification shots quality applied many times I'm sorry it also turns out that we have the tribal opposing half plus G In you can it was founded by thing you can observe plus In that I finally let me observe the domestic properties it also has a U 2 is less than equal to the 3 exceptions so all of these facts follow from something called the Garris Koshy Shrontz inequality and from Gavaskar shots which as I said is just the sort of exotic version of the Commission shots inequality improving using just the coaches shots inequality Is this summer so is something this the enthusiasm of the fans so it is the same as in the I think it was on the verge of knows good question the these so these are universe said and the efforts of function you right after the function bonds and let that From this adds to the complexity of function yes
sir let me explain actually I wide introduced this centered on so that there are many advantages of working on a 100 billion group this makes many averaging arguments a lot easier and you don't ever have to worry about regret the range that things are averaged over I said Now if not I suppose I suppose that is a function on 1 whatsoever then what we do is we embed 1 up to an insider slightly group so that end in sight said modular and crimes but it doesn't actually matter which and pride which choose to just take it to be quite big and so with informed bigger than the 10 K times and say and it's also convenient for technical reasons to to take it bid price and now we define the garrison came along the proof of that on Net city is the norm at the same function regarded as a function on the group this is a slight abuse of notation F is a function on 1 or 2 and but it can also be thought of as a function of this group just by setting it to be 0 outside of the image of Latin betting at normalized by the normal the the interval of outside groups so that was very tactical let me explain to things about it 1st of all it doesn't depend on the chosen primed noting the reason for that is it's it's just the same thing as the average you can also express like this but when the average is over only over To to the case samples for which all of these elements lie side into 1 of 2 ways the same answer this could stall in which the average day is only over X plus meager Dr. H. lying inside 1 of life but other North events is nature's to average over it's kind of a complex subset of 2 sets the case plus 1 at the time I'm not going to dwell too much on the precise distinction between the Nall of on functions up to the end of the normal this group they really the same up to a constant factor you lose you just slightly news some properties like this monotonicity is not quite true you have to include selective factors of 2 things but it more or less the same set some of their ability to find the garrison or similar defined properly what of sequences are because last time I didn't even and I think many people know that I didn't even remind you what not leak repairs in general actually it's good to define these in greater generality them the meeting that would come back in just a 2nd you can never and see what they need to honor it back on the case that we talk about filtration and I'm trying to be consistent in wide use of terminology but I can't guarantee that this is all standard terminology select GPA Group a free filtration on G which I will call G don't it's going to be a sequence of subgroups G. I like to infinity is a nested sequence I'm gene is equal to Chennault contains G 1 contains G 2 this containment symbol it includes the quality case said maybe I should just make them completely clear with those investors sequence all subgroups all and we say that it has class I guess class most analysts they found all of the terms from plus 1 onwards also not trivial so here he is the identity of mental injury I cannot say that it's a filtration just as the 1st 2 terms are equal said Jean or sequence G 1 I forgot the most important action of a pretty filtration this is this is a moment the nested sequence of subgroups and space dissatisfied satisfying but the Commies a set of GI with gj is contained in Shiite plus J for all I'm John why injected 1 at end of the day of his era so let me just remind you of the commentator to Greece's page completed K is the group generated apply the
pairwise commuters it's Agent K agent can homes H & H and K and K and not the same thing as the set of commentators level filtration no there's a particular filtration particularly well known filtration it is the lowest central series filtration so the lowest interest series is defined by I gene just a distinguishes await use the brackets as a gene or the equals G 1 with G & G I plus 1 is the commentator tree with G. I so is actually the maximal filtration because any filtration G commentator GI has to be contained in GI plus 1 so it's actually the maximum space maximal minimal it's that let's go with maximal maximal filtration on GE any other filtration will have GI contained in jail without the brackets just by induction scenario with the minimal filtration and is not absolutely immediate that the low Essential series is a filtration habit that is true and that's easy to prove by induction so this is a filtration by induction 4 so great is as is 1 in a Group is no patent if the lower central series filtration finally clocks the group group is no patient Class S the G 7 plus 1 is equal to the identity II in the low essential serious filtration filtration has class most tests and because the low social series filtration is the minimal filtration agree when he admitted filtration of finite class if it's no presence so that's 1 no present the leaders give a quick example the Heisenberg that we saw last time and she is 1 1 1 on on all said Gene Autry equals to 1 equals G G 2 is the commentator subgroup G cheap which is 1 and 1 1 0 0 off and then G-3 3 is trivial there's no patent of class too so what I mentioned all of this well it turns out that for the purposes of the theory I'm interested in this nothing really special about the lowest interest here is very occasionally below Essential series Miller needs to be discussed but usually Everything was just for arbitary filtration and more it's good to have the flexibility of of voluntary filtration say for example that placed under into intersections with subgroups where is the low interest series need not be so it's a much nicer category for or a while last place to work so polynomial sequence is something that is defined relative to a filtration plant in sequence so the group definition and let's let Jean bullets equals G I I consider its infancy the a filtration and the group of polynomial sequences Pauley says she don't the animal pollinators sequences In PT friends and to so this definition isn't depends on more than just G-8 depends on the filtration that is generated by is the groups generated by sequences of the form a little to the PRI events so there is a sort of basic polynomial sequences well I G I must always lies in a suitably high elements of this filtration so we saw this I gave some examples in the highs group before on Monday allowed in the Eisenberg Group many allowed quadratic terms which sits inside the subgroup cheated Seiji example how 1 1 1 alpha and beta and gamma and squared it's polynomial is in Polly Z G bond energy bullets when she but it's the latter
essential serious filtration on the Heisenberg this is not
absolutely immediate but it's an easy exercise to write this is a product of a more basic sequences like this now again it depends how much time I have but is actually very remarkable theory of these polynomials sequences in the case where the group is no person I should say sorry I I knew what to make this definition when GE is a finite clocks I said she is no better the filtration defined class this just a quite remarkable theory that I want shady some bits of at least due mainly to an Alex on the label but also to Hosston across and this is the idea that there is some different characterizations of what it means to be a polynomial like in the sense that
the alternative characterizations of that his if you remember the include holding yes yes there is some alternative characterizations and the most remarkable and P lies in quality of the said G. but if and only if the you have this derivative property Delta H 1 of those aged 2 to Delta James people of and takes values in she subjective for every choice of H 1 which during the Civil War H 1 of edge in the integers and the control J. greater than 1 where he Delta sub H well it's kind of a multiplicative notion of discrete derivative on this group said Delta sub H In fact that is after kind of endless H the interest so this is some hope this is very natural definition of polynomial the these are to represents a new assignment some have certain derivatives lines and subgroups so this is definitely a nontrivial thermal 1 fact that you made immediately knows about this is that with this definition here is completely not obvious that these P former group that's just not obvious at very non-obvious that is screwed With this definition so in many ways this is the most really this is somehow the natural definition and that is a fair and that will an of sequences are about form and that everything about forms of sequence the new year again this is important that GE is always operation the people of the news yesterday is a filtration of finite class that implies there is no pattern because the low central serious sits beneath sits above the filtration such beneath infiltration and and therefore the filtration eventually terminate said as the Los Angeles area so you can only have a look we have a filtration of finite class in the present case and actually was an example which I don't remember off the top of my head but that if geez not no patient these 2 nations are different said the appointment of sequences satisfy 1 but not the other still other any questions on this before the yet this room contrary to use U.S. I think I am diamonds at the other way around you can have I mean you could have really quite big felt quite flabby filtration is that it is constant for a long time and then stopped taking them In fact the tho was a very trivial filtration which everything in is just the Group G it's not a finite class that if you think you have to work very well he was called to the students whatever label things like that the filtration properties considered slightly with the basic idea and I J Atlasjet minus 1 but this this turns out to be the easiest way to with thinks aftermath with the board was vegetable to get this 1 Donegan and that's it this is possible so this is a very algebraic theory on which I may give some hints of if I have time is not it's not that easy to prove these facts many have 6 election so this is biologist in the setting of no potent groups but I'm exclusively going to be interested in a much more restricted setting them up this will only be interested in and I think this is not very happy with this piece of nomenclature but let me say this the proper filtration maybe I should call it a leaf filtration that's a better term but let's stick with proper filtration so these are ones in which everything involved is a closed and simply connected L'Equipe said will simply connected group in which she is a simple connected and the group and each GI is a close connected itself said the Heisenberg example that I showed you before it's definitely have had time to think Heisenberg example I should let me give you another example this is a pretty trivial example but I'll mention it again later and another example would be To take GET off and Gene ought the cost you 1 equals GST "quotation mark time and then G as plus 1 and will hire sentenced to just be trivial so that's what I'd call it a quite a flabby sort filtration is definitely not a minimal 1 and an exercise 1 that's just follows from the definition they would appear in a traditional polynomials PT from to all of degree that most so EG & peer Aventis Alpha and the answer this will be polynomial sequences in the sense of fun my definition that in a more general sense so it really is a generalization all
familiar nations so now need to introduce if I want to about no sequences I want to introduce and last and of would the 1 running the site maybe I should say but I think it's it's instructive to see the general definition of these things but personally I I usually just think about examples on the Heisenberg group which have a lot of the general structure already although not all of it so lattices and more functions so alas it is just a discreet and compact subgroups gamma less to G is paying lattice if it is discreet come Co ,comma so we've seen already an example in the Heisenberg the G 1 1 1 said Said said in the so there may not be a lattice injury there examples in dimensions 7 I think of class to nobody groups admitting their losses but that's totally wrong for me the I want to give you very much the sense that this is not even argues words like simply connected I I'm not doing any kind of topology and this is really algebra what should think of these things is like vector spaces they have no interest in topological content ready I'll make that clearer because I'm going to mention the LEA algebra saved being a simply connected Nova to Lee group means that you want the exponential not from the algebra is Ajami amorphous and so they really are algebraic objects I would say so it's possible that she admits molasses but I simply won't care about such cheese in this theory that has never come up I'm always going to see if we say that the filtration g but please rational but what if gamma basically forms the lapses in each element but sequence I will only ever be interested in rational filtration but again there there sort of Fort Ord call soft results and by softened it mean easy their statements Jews emulsion and along the lines of that but any lattice will be rational with respect to the lowest central series believe us there but again it's irrelevant to us every situation that we given it will somehow be clear I know that these cities these things hold on I'm not interested in pathologies any questions about so that's and I can define automorphic function and for me and automorphic function "quotation mark why smooth automorphic function it is some 5 in Infiniti's G since the is liturgy is a meaningful notion she is supposed to be in the group satisfying that fly all Gamma G equals 5 G for organic can come so now I can define precisely what no sequences In September edition they know sequence of Class S is a function but kind abandon this fire appeared well well where all of the objects written them off as above P is polynomial With respect to some filtration there was some proper filtration and she almost and I'm which is rational rational with respect some molasses and gamma and some smooth Donald Moffat functions so there are lots of formal definition of what a no sequence it's it's a bit more complicated involving this additional structure of a filtration and which must be rational with respect to a leftist since many now is a good time to make some remarks about the site I don't necessarily think that this particular definition will last forever Moore and his wife exists I taught about no sequences being the generalizations of additive characters which are each of the 2 fees so the there's accounts a very special type of no sequence and here allowing any smooth function and the reason for that is that we don't know which is somehow the natural subclasses smooth functions fire that we should be considering so maybe maybe 1 should take bike and functions of some operations on June 1 ,comma I don't know I don't know if this seems to be no persuasive reason to select 1 class at the moment so this definition is potentially subject to revision in the future but for now it's it's very workable 9 the Gallas in the statement of the investor for the Council devoted notion of complexity and without that it has no content because these sequences can be just any for these can take essentially arbitrary values on arbitrarily long intervals states to make make it very useful we have to have combination of complexity
this year I saw I want to weaken definition because I want to give a complete statement of the inverse of the Council on but please complexity issues in this subject they're always annoying and then never conceptually difficult so there's something I always just put it what town myself words would just put it in an appendix to undertake the kind of always work out in the end a bit like couldn't if anybody's done linear algebra Over Q where you have to keep track of the heights of rationalizing using these things always tedious but never really difficult was sometimes with aggressive was the situations where it has actually very important to keep track of 6 but for me there never be conceptually difficult said To make the theory quantitatively useful but maybe before saying this I should also say that the nation no sequence 1st erasing work of bogus and Hosten cross in ergodic theory and then nation was a little bit different to this in that they enabling theory it is somehow unnatural to just use a continuous function fine and they didn't use polynomial sequences the use of special type of Parliament of sequence just the linear sequence the Nikkei's sequences arises only in this quantitative theory for reasons that all mention is made of the quantitative useful we need to know we need measures of 2 things really have the complexity all of the filtration with respect to Gatlin and and also has smooth the function fires the smoothness all 5 but to quantify those nations when you have to choose a basis with respect to which measuring these things and the way to do that is to pass to the LEA algebra and so would work in the algebra Kerr-McGee agreed to run into trouble here and you need to make sure that does not look like a fly yeah as well as in the algebra G I'm why myself and not Matsuda familiar with Lee theory I don't want to go over huge amounts of the theory but what I recommend if you're not familiar with adjusting the Heisenberg case figure out what the algebra is it's a three-dimensional vector space with a of bracket operation on it so the thing is for the simply connected since she is still potent simply connected with and we have homomorphism permeable residents not homomorphisms homemade mortars and between the algebra called the exponential map on the logarithmic and as I say this remains were doing algebra and and definitely not only hunt supporters so we algebra now the unimpressive vector space I can just pick a basis for it complexity I the complexity is measured can only be measured with respect to a particular choice of bases got choose a basis Kelly for the algebra G plans now it's convenient not to choose an arbitary basis but to choose 1 that is adapted I would say that currently is adapted to the filtration X 1 up to x sub I did because I want to start from the other end the Act X sub didn't she -minus she I plus 1 top to sub thingy is the basis for the algebra of GIA which is defined to be the law chief In the filtration start adapted basis is 1 that's good for computing with respect to filtration the ones would make such a choice we can quantify those 2 nations above so I can I give you the definition of both of those things the 3 questions on that so I should have time to just finish this before In the end of lecture all of the area the definition how we say that's the complexity which I would write harsh stop of the filtration and gamma because that's the thing that depends on it is at most and if there is and it space to be integer if the following matches so basically the at the idea is that I want 1 of the interesting things about higher than the filtration and the lattice interact to create it find in terms of how and it turns out that these interesting things so 1st of all 1 of the structure constants so the structure constants described lead racket In terms of the spaces so those are the I'm J K they satisfy their work most on and the rules and rational and I J K lies inside the integers the system there is this result of motion that says that a would only
admits lattice civil worthless if and only if the structure constants all rational so this is a quantitative version of g missing Alaska I am and I find it convenient when writing notes on this to introduce just 1 slightly technical extra conditional 10 1 amendment and so this is just purely technical but I would give things completely an images from moments of exactly what it was yes the value of there are at most and the values of Jr such it was not clear if not not trivial I included that just because it it makes these nations behave better with respect to certain functorial properties groups like taking direct products From that's quite convenient they just ignore that and then secondly it is the assertion that and time the integer lattice in the basis B is contained within law of the losses ,comma which is contained within 1 over and gelatinous on the basis that so this is another statement
about how the lattice sits inside hum berries with respect to the spaces so long gamma is just the image under the logarithm map of all the elements of the discrete creep ,comma is actually not necessarily Latin is not necessarily a subgroup actions so you could be careful with these Lee operations they don't not definitely not homomorphism so this is not it's a lattice up the finite index but it's not close and addition necessarily and fat for the Heisenberg it's no so that's a definition of complexity and then I have to define how news fight is well that's fairly routine and it turns out that get filtration standing in the dining room I thought would be 10 minutes as I can I think will make it he'll spend the 1st few minutes just recording contribute efficiently seemingly not this France where I think I can manage it in 6 minutes so we're going to quantify this smoothness using some basically what is the least well no directive norms and we define the smoothness norms smoothness and songwriter W and can be this will be the supreme all from basically all the relatives with respect to the basis so that's an enterprise find in Phinazee so whether Supreme is overruled choices all of y 1 up to buy and crimes and overall derivatives of order at most so it's just the biggest any derivative convenient derivative here is in the usual Liotta stands the when and the X 5 is the derivative 0 and Steve ITT equals 0 of 5 sometimes effects all expert T X so this is a standard definition and so I don't know if there are any genuine experts in the audience or and this is being videoed and people watch these things but if said I should like to remark that these notions of complexity a rather different to the ones that I used in papers by telling myself where we talk about it Lipschitz norms on fire which require once put metric on gene modular garment I'm not comes the conclusion I'm not sure I think Terry came to this conclusion he is a guarantee that says smoothness norms are then the right way to proceed and invited other people who work with quantitative they could distribution results such as this papers of buying Seidler Margolis and thanked test they use the smoothest norms as well 1 should not be working in the Lipschitz category smoothness is correct and so let me finish then by just stating precisely the investor the the guy was so it's on my basically states that before but so supposed but back in the guy was enormous at least a 2 of the land there is no sequence kind and which is 5 years well filtration has class at most came on this 1 so when
she told her has class most came on this 1 and with respect to a suitable basis for a suitable basis Kelly be adapted to GE but we have but the complexity with respectively all the filtration is binding only in terms of Delta and care so I write this is bigger down 1
and the smoothness of fires also cites rebounded Hi so the smoothest find for any and finally the
course important beds and at correlates with for some Delta crimes "quotation mark greater than 0 in a way that depends on Delta and came so you can seem somehow why is is not normally stated formally it's a bit of an effort that is the the gist of it is what I explained at the beginning so you correlate with no sequence of finite complexity so next time I will look at the Comverse the interest and for the cows norms and you'll be pleased to hear that although these notions of complexity off in the background and would be important if you wanted to do everything rigorously should not be I'm mentioning them explicitly largest say that they can be bonded so that's it for today this we should go make haste among them


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