1/6 Nilsequences
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Title 
1/6 Nilsequences

Title of Series  
Number of Parts 
18

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2014

Language 
English

Content Metadata
Subject Area  
Abstract 
Classical Fourier analysis has found many uses in additive number theory. However, while it is welladapted 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 3term arithmetic progressions are contained in A, then Fourier analysis is useful, but if one wishes to count 4term 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 «higherorder Fourier analysis». The talks will be about this theory.

00:00
Point (geometry)
Cue sports
Divisor
State of matter
Multiplication sign
Similarity (geometry)
Prime number
Rule of inference
Theory
Asymptote
Power (physics)
Subset
Prime ideal
Goodness of fit
Manysorted logic
Term (mathematics)
Cuboid
Integer
Theorem
Hadamard, Jacques
Subtraction
Alpha (investment)
Physical system
Series (mathematics)
Spacetime
Element (mathematics)
Mathematical analysis
Set (mathematics)
Sequence
Bilinear form
General relativity
Number theory
Hadamard matrix
Computer animation
Vector space
Network topology
Order (biology)
Linearization
Right angle
Object (grammar)
Arithmetic progression
08:34
Observational study
Multiplication sign
Equaliser (mathematics)
Propositional formula
Theory
Order theory
Expected value
Manysorted logic
Positional notation
Hardy space
Term (mathematics)
Average
Wellformed formula
Square number
Subtraction
Combinatorics
Orthogonality
Alpha (investment)
Social class
Addition
Spacetime
Theory of relativity
Reflection (mathematics)
Exponentiation
Element (mathematics)
Mathematical analysis
Algebraic structure
Set (mathematics)
Fourier transform
Complex number
Sequence
Functional (mathematics)
Local Group
Convolution
Proof theory
Category of being
Summation
Arithmetic mean
Number theory
Order (biology)
Equation
Normal (geometry)
Arithmetic progression
Identical particles
25:30
Multiplication sign
Direction (geometry)
Range (statistics)
Propositional formula
Parameter (computer programming)
Mereology
Weight
Dreiecksungleichung
Roundness (object)
Manysorted logic
Positional notation
Square number
Keilförmige Anordnung
Theory of relativity
Spacetime
Rational number
Moment (mathematics)
Perturbation theory
Functional (mathematics)
Bilinear form
Maxima and minima
Category of being
Proof theory
Arithmetic mean
Ergodentheorie
Oval
Normal (geometry)
Configuration space
Right angle
Uniform space
Arithmetic progression
Point (geometry)
Divisor
Observational study
Routing
Inequality (mathematics)
Event horizon
Theory
Frequency
Goodness of fit
Population density
Lecture/Conference
Wellformed formula
Average
Analytic number theory
Lie group
Analytic continuation
Subtraction
Orthogonality
Standard deviation
Exponentiation
Expression
Element (mathematics)
Fourier transform
Set (mathematics)
Approximation
Local Group
Calculation
Summation
Number theory
Network topology
42:26
Calculation
Roundness (object)
Number theory
Lecture/Conference
Element (mathematics)
Square number
Arithmetic progression
43:34
Point (geometry)
Randomization
State of matter
Multiplication sign
Propositional formula
Parameter (computer programming)
Inequality (mathematics)
Mereology
Glatte Funktion
Fraction (mathematics)
Root
Lecture/Conference
Natural number
Square number
Lie group
Alpha (investment)
Counterexample
Theory of relativity
Forcing (mathematics)
Moment (mathematics)
Exponentiation
Variance
Line (geometry)
Set (mathematics)
Approximation
Vector potential
Summation
Number theory
Arithmetic progression
Identical particles
51:23
Complex (psychology)
Beta function
Multiplication sign
Group theory
Mereology
Glatte Funktion
Matrix (mathematics)
Square number
Process (computing)
Gamma function
Linear regression
Moment (mathematics)
Sequence
Functional (mathematics)
Automorphism
Connected space
Arithmetic mean
Lattice (order)
Right angle
Arithmetic progression
Identical particles
Resultant
Point (geometry)
Classical physics
Sequel
Presentation of a group
Observational study
Real number
Process capability index
Inequality (mathematics)
Event horizon
Quadratic equation
Frequency
Lecture/Conference
Term (mathematics)
Alpha (investment)
Series (mathematics)
Multiplication
PoissonKlammer
Exponentiation
Volume (thermodynamics)
Set (mathematics)
Euler angles
Cartesian coordinate system
Local Group
Subgroup
Field extension
Summation
Number theory
Network topology
Object (grammar)
1:04:48
Classical physics
Pulse (signal processing)
Group action
Interior (topology)
Multiplication sign
Similarity (geometry)
Water vapor
Mereology
Food energy
Theory
Pi
Mathematics
Matrix (mathematics)
Manysorted logic
Lecture/Conference
Meeting/Interview
Natural number
Reduction of order
Integer
Subtraction
Field (mathematics)
Fundamentalbereich
Gamma function
PoissonKlammer
Functional (mathematics)
Connected space
Calculation
Lattice (order)
Cost curve
Right angle
Figurate number
Fundamental theorem of algebra
1:11:30
Point (geometry)
Complex (psychology)
State of matter
Multiplication sign
Theory
Quadratic equation
Matrix (mathematics)
Manysorted logic
Meeting/Interview
Lecture/Conference
Average
Spacetime
Gamma function
PoissonKlammer
Expression
Functional (mathematics)
Bilinear form
Sequence
Automorphism
Local Group
Category of being
Thermal radiation
Phase transition
Linearization
Charge carrier
Normal (geometry)
Arithmetic progression
Family
1:22:51
Point (geometry)
Axiom of choice
State of matter
Multiplication sign
Mereology
Inequality (mathematics)
Prime number
Power (physics)
Prime ideal
Explosion
Roundness (object)
Manysorted logic
Strategy game
Lecture/Conference
Meeting/Interview
Term (mathematics)
Average
Social class
Theory of relativity
Algebraic structure
Nominal number
Cartesian coordinate system
Sequence
Bilinear form
Functional (mathematics)
Logic
Tower
Hausdorff dimension
Linearization
Normal (geometry)
Arithmetic progression
00:06
the wound to the good good
00:12
good a half and the the the the Internet send them to be going on you I if the union a and yes was with the invitation 1st thing is any chalk up that I I can't see any whatsoever always 6 years so I'm going to give 6 lectures and um this first one will be quite introductory ahead of my abilities offended by me going over some fairly basic facts at times other let me start by explaining what the lecture series is about so the title is still sequences but I think a better title might have been higher world of Fourier analysis I said I'm going to be looking at various questions often and usually an additive and combinatorial number theory when the traditional techniques on involved Fourier analysis and I'll explain that a little bit about and that these techniques only go up to a certain point and then made it work and I much more recently this this idea high order for analysis has been interesting no sequences all the characters of higher order for analysis so quite often give maybe 1 or 2 hour lectures on this topic and it's must have the opportunity to give the longest series of lectures because I can go into some more details about these objects in their own right and this as an interesting sort of algebraic aspects to the theory that I want to add to well on and a bit more detail rather than normally would so we begin by telling you what the what some of the questions that we might be interested in answering set and goal said Harris 3 pheromones only 2 theorems that this theory can be used to prove that I will be be giving all of the details of either of them but hopefully they provide some motivation for wanting to learn about these objects so the 1st goal and is to understand linear patterns in the prime numbers so 1 example would be for example this is a fair and this is a thermos Terrence term myself Montana glow from me 2011 said state 1 particular case the there that the number of KTM arithmetical progressions of Prime so the state crimes less than X which is equally spaced previous asymptotic play I'm a constant times next squared I have a lot of the context but the video just pause to explain why that's a natural Olsen and the number of hours charismatic progressions of the vintages less than Texas essentially EC squared you can choose the 1st time Annex ways and then the common difference in something like X ways and then the probability that a given number is prime is 1 of a log annexed by the prime number there so somehow this term here is not terribly surprising and then there's an arithmetical factor here and which deals with the fact that prime numbers usually order and don't tend to be divisible by 3 so is absent offered this for some explicit to seek a great accessory and actually all Allstate later on a more general theory this is not just a thermal about arithmetical progressions and they're much more general patterns prime Minister and can be considered In fact you can numb apart from instant degenerate cases apart from the the general cases um we can essentially can how often all but 3 sets of linear forms of integers takes pride value so I want founder and 2 society here a rule Prime where there is someone out subsides he some linear forms in the variables and on the vector and ranges over some sort of open settled a box or something like that and ranges from myself unfortunately the degenerate cases include most of the really famous open problems like the twin prime conjecture and go back conjecture and but somehow almost any other systems is covered by this there so that's 1 main motivating goal and the other 1 is questions related to what's called Summer 87 is system ready serum is a very famous theorem about not about a specific set like the primes but about all the trees I suppose we suppose that a is a subset of the 1st then introduced I suppose it has a certain size powerful than similarities serum states that the end is big enough In terms of Alpha and Kate then a war contain K distinct elements in an arithmetic progression and assesses is
08:42
that there has been proven many times szemerédi's original proof was ,comma tutorial and there's a very well known and very beautiful proof using ergodic but that it was Timothy Gowers who I understand it in Paris today and speaking at 5 PM and was interested Timothy Gallas proof of this which some high kicked off the the whole subject of higher order for analysis Selby said a few things about still talk about Gowers proved Gamez main motivation I in coming up with this proof I think Woods was to get sensible binds for this quantity pleaded previously there being no respectable binds the Heiberg this quantity had to be in gallons finds them but I I'm not been focusing much on that on the underlying structures and I may talk about sort of curious refinement of this that really has become a decline ever get that back home which tool is recommended for that UK but it needed back backed imminently status as long as wouldn't mind looking at them set a curious sort of addendum if you like and and this this needs much more of the theory that I'm going to talk about they're actually there is some common difference some common different stadium and such that a contains at least she quite a lot of arithmetical versions of this particular common difference so leased office the case of little 1 times and progressions of common difference however little 1 means a quantity that tends zeros and sensitivity cake was 3 and 4 and Netshitenzhe either that's false the cake was fired and higher so this is best possible if you just take a random set I just pick elements from 1 of 2 and just randomly with probability Alpha than you'd expect the number of progressions with common difference deed be exactly that the many bodies of this is a cure is a much deeper facts that has to do with some subtle properties of the notice and groups of class to which I'm going to be discussing reflections says Monday's to Theron's will be motivation but really I want to talk just more generally about this thing called higher order for analysis so that's actually begin properly so why why why no sequences or higher order for analysis so the answer as I said before is that there are some problems that have been that can be handled by traditional for analysis so some questions as surrendered race at let me ask you questions about security questions to additive number theory questions however 1 can ask about Justin arbitary set a contained in the 1st and injures so Question 1 and all the solutions to the the equation explicit White was that in them so that's a very basic type of combinatorial additive number theory questioning people would ask and Question 2 much more related to the 2 questions I've put on the board although fourterm progressions insider upon the full term harassment X X plus the explicity and plus 3 in and what I would convince you of it is that tools solve very classical tools of Fourier analysis I can help you solve Question 1 but that there are than not sufficient for studying question 2 the answer this it's has traditionally been the motivation for introducing a higher order theory sir now the since study question 1 and other related questions we use classical for analysis I'm well quit exponential sums it has many different names so it's also known as for analysis and depending on exactly how you apply it and some people would call a hardy Isherwood method resume different facets of the classical for analysis so the exponential some attached to any place defined as follows the define s sub a of the sea To be the average and the analyst at the end of the characteristic function of a sometimes each the 2 the session what is that so the 2 pieces of notation here I'm going to use quite a lot so that may explain the problem here once OK and it's 1 of ends in May and 0 and is not in and a slightly curious space but very useful tradition or recent work and additive
17:02
number theory is that we use the expectation symbol to mean average said it expected value and that unless an accord to end it's the same thing as 1 over and times the sum of analysts and is a careful use the sideboards parents the started so if you want to compare How many solutions if you got 2 different sets a and B. let's compare how many solutions to expose White was that they have it turns out that we need to do this compare an exponential sums In a sense that's a trivial statement because the exponential some determines the set I have but I mean in a slightly more I mean it slightly differently so that misstated a proposition as I suppose I have 2 sets inside 1 of 2 and then let's can't pin number of solutions to X X plus White was set inside a so the number of exwife which X Y and explores why lies in any and as compared to the number of Paris for which Texas wineries plus wireline B but look at the difference between them I so the conclusion this yeah if the exponential sums over a and B a closer than these 2 quantities of clothes and so this is bounded by 3 Delta and squared when Delta since the Supreme overseas have more theater minor success anything so the weather like to think about this days exponential sums it all up but good enough to war counting solutions to these things exponential sums suffice for studying X Y and X plus and it's true that proposition just to show you what I mean by Fourier analysis the key is that there is a formula for the number of summing Triple X Y plus wise in terms of the exponential something so proof we have the formula and that the number of text wine which X Y and explores why lies in the way it is virtually equal to it's and squared 2 times and Cup's time and the integral that essentially the Cuba the exponential now when I put this formula from well let me 1st markets it's not hard to prove this follows by hand from the orthogonal relations so if you just as if he substituted in the definition of the exponential some essay feature so just as a consequence of the orthogonal the relations that turns into the 2 Paul the 2 animated feature is basically wants 1 of them is 0 and 0 . com is not so that's a fun exercise to expand the license and check and they're more conceptual ways to arrive at the full really was going on is that the number of solutions exwife wineries pretty much of the triple combination of of 1 a with itself and it's probably most people know the convolution becomes most occasion when he passed to the Fourier transform and that's what's happened here and so that's how you to rival the former perhaps assist similarly for the um and we also notes that the integral they'll to mean and this exponential some again by the orthogonal the relations is just 10 times the size of a which is less than a week when square and again similarly could be so this is just another possible identity this defacto Q a many of starting something says they title again that toppled with this wasn't doing this so the only question so far and was very quiet on the 1 who just came back from Los Angeles and is feeling sleepy should all try and keep me awake that was over yeah I think so it's 7 wired to refer to my notes it's to do with the way have normalized the exponential some years but yes exactly so the normalization disposed to their chosen so that expenditure some is always between 0 and 1 1 nite this note just a simple equality of complex numbers said mode said what W elastic to bond then the size of and that's where it Z minus W square W can be expanded as that squared said minus W bomb plus another term instead minus W Z plus W W "quotation mark I'm sure
25:34
it's not sure that it to will use the summit to you in 1 of the people you work is not a stupid question elicited dumb I must've gotten normalization they're wrong said if you and I think it should just be the size of a actually the future not only by the way divided by and square exactly here because I normalized its status as this is the most 1 over on here thank you yes this is I mean this is just hour to break calculation this is bounded by homogenous so modulus of this his bounded by 3 times month and my stomach yeah so now we're going to apply this with this formula here so the difference between these 2 sides is Cuche times the integral from 1 2 0 S feature if squared feature minus the same with the I'm so by the triangle inequality and by this something weird about this so I want to be but the use of yeah I II and Kayseri and there is no that I shouldn't know it will work anomalies electrified events I forgot to add yeah I know I need a statement here with a win over the MoD said squared plus a W square tales about which would certainly be true the so this is bounded by and cubes time some that is the supreme on the feature of modern feature minus SPC said sometimes the integral from 1 2 0 on what s feature squared plus month as the squared deceased which hopefully so good over and the article on the right so this gives me 1 ran essentially amounts an end squared and the should be dealt with all check what so lost a factor to secessionist 6 Delta and square Strange that's wrong I don't normally use notes and at the computer but no journalist Donna a case so basically there's a routine argument for deducing the statement just using the Fourier transform and we need is the orthogonal organizer relation which gives you formula um further expression we want together with possible that entity he said the people of 2 of you the most of was will build the I'm in it a case of I at this evident this to be a more z squared off in an animal doubly squared across is certainly most and psychic recently replaces by 2 years but I 302 invites River to work CSU right location location shots so I can rescue the exact statement I had expected again but it's not quite as simple as I made home and the fact that the MoD said W. less than 1 is actually taking relevant to the to history for any this is 1 of those on the Battle of the Nile so is it minds might watch my W along OK and I think thank you Secretary said everything is now I believe correct on that board and in features history the statement claimed is also correct but right now the main point that I was space to be just that that was very severe classical exercise and the main point is that this fails fall more exotic configuration he said is a proposition in a negative proposition about 410 progression and there exists sets a and B inside 1 2 and I both of some size Delta and now with the property that their exponential some place says some the feature of modern the 3rd sp it is little 1 Cyrillic place and but they have quite different kinds of 410 progression the number of forts impressions the number of X and the next explores the explores explosive 3 in a minus the corresponding carried the baby is a leased some positive constant Delta primed 100 square so this is saying that In a strong sense that exponential someone not a good enough tools study Fortum directions we have 2 sets
33:59
use exponential sums paid essentially the same but when you try and have fought progressions that behave differently exponential sums cannot handle Fulton aggression so in the sense that an infinitesimal a tiny perturbations exponential some can have a huge preservation 2 the number of digressions and the example here is kind of motivates the whole theory in a way this example is a guest what is due to Gowers but it was some In very related concepts of ergodic theory was too is due to Furstenberg and Vice amendment going to describe a slight variant on the examples so just take ACP take a severe Rondon said In the evidence to Delta I should say this is going to be a sketch and sketch proof rather than example minute described the example trees on the sketch group to reach let's fix stuff I'm don't the reason any sketches that somehow you did it's quite easier thing to convince you that something does go wrong the exponential sums I'm not a sufficient tools handle fulltime progressions is that she kind of hard to rigorously prove that they're not that somehow there's not an interesting thing to do light weight wouldn't need to rigorously the once we convinced that they're not they're not going work with any to rigorously proof that it works or not going to and acidic be around said density Delta and I guess I mean by that just pick elements to lie a independently with probability .period and then something that makes some claims that I would approve of say something about related but mean number them so what's the exponential some a was basically the exponential Over the whole interval 1 of the when but are selected each point and with probability Delta said Delta times that and with high probability this will be true uniformity in the so with high probability so rigorous justification of that's not totally trivial he could use something like the 2nd method exercise and using 2nd moment method for example or large a a large deviation back and it's not taking trivial because you've got to there's a whole a continuous range of features that you need to worry about infinitely many events but if the service is prime declares the essentially free similar events so you couldn't just using union and overlays and finally many events and that's a very believable back and I was beginning to be so be it said the Peter be set the wall and listened and for which the fractional part of and squared routes to lies between 0 and Delta Nahid say here that's the fractional part of a team and there's nothing really special about square it 2 and it's just you want to choose something that's not close to a rational so we use about refer to is poor approximation by rational so I claim that the the exponential solid is the same moral lessons for rounds for this the 2 it is the same claimed the knife around inside species is also Delta times the exponential some of the whole interval about wedges realize I had lapsed into this is a standard analytic number theory notation that may not be familiar to everybody guilty pleas e the twopart II times always plus little I want Nice visits that's natural I mean this said B I mean if you draw is it looks a bit like a random said you I defy you to distinguish if I just breakdown of the border random 100 elements and this set you probably wouldn't see the difference between but the proof of this statement is it is a bit delicate so what you need to use its files inequality for a quadratic exponential sums so this is a so an inequality sums averages in the form of year Our France where KlausPeter and has a standard sort of thing in analytic number theory and that the basic factors that if highly a rational than this is suspended quite nontrivial the trivial binders 1 but you need a bit more than that as well you need to somehow smooth nightly and you need to smooth out this interval 0 up to Delta and expanded as a Fourier transform In insure you need a bit of technique to rigorously professor so plus smooth approximation to the characteristic function of the Delta so this is a tricky exercises In pretty standard analytic number theory techniques OK so the exponential sums because they both won that they're essentially the same and hence 1 plus 2 implies that the sort of theater "quotation mark this statement said that special from the know what about the
42:27
Fort enlargement progressions so how many Fortum progressions does have to but what it is relatively easy calculation said the number of Fortum progressions between 1 and ends and the number of X and which X X plus the exports to India next plus 3 days are listening and its roughly onesixth Sundance Square so that's an easy exercise society just integrating over twodimensional demand and then because aides round of what you fix 1 of those progression provided the elements a distinct let's assume they'll it will
43:36
line Saturday with probability
43:37
Alpha to the fore said Delta 4 so the probability it lies in any of the roughly doses of 4 of the 6 thousand square Salemme formerly state answers . 3 the number of Fortum progressions and is and those of Florida 6 times and squares supposedly with high probability so guess that's something I haven't actually proved if you think think about it the argument I just said I was competing the expected number of Fortum
44:37
progression today that's not the same as showing that this you need also to know that it concentrates around me but again some 2nd moment method would give us so you can see why sketching This is a collection of files that are all very believable but to Prudence hard work and it would be somewhat pointless because it's all it's all negative those things all just a counterexample let me right here now comes the interesting thing which is the number of Fortum progressions in and the reason this is interesting is that there is a certain identity said observed the identity that X squared minus 3 times explicity square plus 3 times X plus 2 D squared m is equal to X plus 3 square says that value natural identity and what this tells you let me defined as to be the following sentence as is susceptible and less than 2 an a hour for which 370 Delta is less than or equal to the fractional part square roots to his left with 470 Delta that relation tells you that if if x x bustier and exposed to the lion s then automatically X 3 lies in an explores threeday lies inside and so it follows easily from his identity and the triangular quality the 7 is 1 plus 3 plus 3 is utmost movies we should adopt to is contained inside a all year said as is a smaller 7 all that would typically have about oneseventh the size of a fire said and the number of forts and progressions and it is at least the number of freedom progression in now to evaluate a number of threesome progressions this is what this lengthy exercise 1 again I'm going to tell you How long would do that in the it's of the year In this long enough you're asian people exactly so that the next bus readers as does not lie in that Lisonbee for this year should be come well attended the answer has some the same exponential some of the random stuff this potential some address is close to and the exponential sum of around instead of the same size it's a random set of size 7 Delta and 7 infected I wanted to prove that you need to do the same argument again and you need to use balls inequality smooth approximation and so on as in Part 2 I mean but then by variances the proposition I proved over here so this wasn't about freedom progressions but it was about of closely related I think X Y and X plus wine that will tell you that the number of 310 progressions Annette is actually pretty much the same as the number of threesomes progressions in a random set of this size set by an argument similar to the previous proposition when I think about it did occur to me that perhaps it would be natural to deal with freedom progressions in the previous proposition Vietnamese and indeed that said it would give the impression that I am obsessed with only arithmetical aggressions because these methods a more general and that's that trivial variant of that proposition it follows that the number 3 simple questions and answers is roughly the number of Sweden progression and around and set aside Delta and 7 and then by similar arguments to this 1 I can't say exactly what but it's so that's roughly Delta over 7 accused timed 1 course and square at the point now is that this behave like Delta where is this behave like doubts as the force set list of Delta is small enough these will be quite different quantities should said that Delta be sufficiently small small
51:24
say yes that that the number of farms the number of forts and regressions in b is basically dealt acute times and squared was in a it's essentially doses of 4 times and square so the result follows as the Delta accused over whatever this is 4 times 7 q doping much bigger than them Delta's the Florida 6 4 small dogs and if you wanted to inflict more punishment on yourself you could even can precisely how many 410 progressions the right in the with some more smooth cutoffs and applications evolves inequality there the main point I want to get from this 1 of the 2 main points this fact that exponential sums all good for some problems but not for others and then there's just this little hints that something else is going on for these more complicated problems that's not it's not a complete mess this is very far from an arbitary sent so there's some kind of quadratic behavior going on rather than linear behavior it is right that was quite a shock the whole of the audience and the key point key points will 1st of all on additive characters he said the 2 Part II these and other good for some questions but not all and then the other point is that we saw hints of some higherorder behavior in this case quadratic k and many questions on anything so far yes the wouldn't know sequences so what is in those sequel so this is going to be a little bit of a league for what I've just said I'm unwilling to really see the connection I'm lecturers sons was going to tell you what another additive character is again so that is a character and to the 2 Paul II fees and can be written as the rise in a strange way so I can write it is fine old appear that I pity From his Entergy How is this the linear is given by appeared on this the surrender and she the real numbers off and is again a periodic function on gene where galleries said so what that means is that gamma of 5 have explicit gonna is equal to 5 x 4 x injury and gather in so not character we know those things quite well I know sequences just it's a generalization of this in which I replaced each of these objects by somewhat more general on Saturday and no sequence of cloth that is a generalization of this in which while he from sets gene is there another tree pollen a amount definition to begin later and she is no longer an abelian groups such as but it's just an arbitrary no present said she is an Estep no patent and the group again I'll be recalling the definitions and and will simply assuming that simply connected on an
58:08
ungiven example in just a moment and then fly it is well perhaps slightly controversial I like to use would automorphic considered periodic once it's not billion so far is Cameron's Moffat whereas Galilee contained in a lattice Escobar's volume said yes I should say could ease the exactly same here fiery and 5 from G to see is a complex value automorphic functions away Gamez alas is also moving means that 5 ,comma X is equal to 5 x 4 ,comma and become so when the group is not necessarily a billion we write the group operations multiplicative late as usual so you can see that this is it To the direct generalization a special case of this it's the editor characters but my job is to convince you that said Elise as far as we understand this phenomenon that these are the natural generalizations of the additive characters these are going to be a bicycle objects study start I will recall this definition of few times over the lectures and I'm also be giving some examples of that it is of course it's a pretty crucial definition haven't said 1 upon and I have said little people no 1 and present repairs so I want to give now an example and this is the example I personally invariably end up playing with 1 of thinking about these things because some highly being case is misleading and then large militant groups are unpleasant In general get in a real mess with commentators sometimes can be quite distressing so the Heisenberg example it's not going to talk about so just fill in the blanks that so he will take GE To be this group of 3 by 3 major cities with the usual matrix multiplication so this is the twostep no potent and simply connected the group cities don't know patient means that commentators border 3 vanish so tax brackets 1 Brackett said is equal to the identity for a tax Weiss said To some extent is arbitrary which way defined accommodated I'll be taking my so let me take To be the standard license to 1 needs to define and no sequins coming from the Senate to give an example of a polynomial Mt what I can do that without going into the general definition so event equals 1 1 1 alpha and beta and 0 and P Peter older it's 1 1 1 alpha and beta and gamma and square debates pollen maps again or organza general definition Lay said that the key point is that all of the terms of the green tea are in the desalinated below Essential series allout said .period please that's In terms of degrees the all in the DEC lower central serious subgroup so you couldn't have a quadratic terms just above at that time he said suharto define automorphic function on GE so there are several ways of constructing automorphic functions find GTC so you all tell you come in this classical kind of way so 1 example of to fire chief it's just the sum ever gonna and gamut of sigh of chemistry so you just take a fixed function PCI and translated around by gamma provided that size some I'm completely supported let's say function on tree I should say I want my
1:04:51
Wafiq functions to be smooth that's important said and so yes you could choose your favorite compactly supported functional on GE and translates it like this get automorphic function and in the notes I have a calculation were show that by choosing judiciously you actually to get the Classical Theater Jacoby the to function this way I'm about to do that calculation here because it's actually 90 irrelevant to the theory that but I think a red herring said this summer under water from the French word for that is it's a connection to a different part of mathematics that is seems to be totally irrelevant and quite specific to this case G so the 2 functions for example the Jacoby the 2 functions arises In this way the military to function itself is not an awesome function but get this uneven dosages proceed tightly by hand which some higher I'd buy field is a bit more intuition into certain aspects of this the 2nd method similar bruteforce method you just define any function you like on a fundamental domain for and energy and the action of gamma so that after the Kenny Smith function on a fundamental domain for a gamma watching and then defined 5 G to asphalt pirate where pies the map into the fundamental to the Islanders know what happens at the edge of the fundamentals of majors for this discussion so let's pretend that this is need to make makes sense sort of almost everywhere is not much and so you can just do a computation here to get some sense of the the flavor of these cost function that I want to do that computation briefly questions on that so computer let's pick a natural fundamental make and figure out what that not pie in the set by choosing them so let's let that this matrix 1 1 1 X Y Z and then we can choose and gamma equals 1 . 1 1 K L 10 in the lattice become such that but gamma times GE has all its entries between 0 1 so instead of all major cities with their entries between 0 1 is a fundamental domain for government G and let's see exactly how can do that indeed gamma times G please 1 1 1 K plus acts L plus Y and struggled to multiply this is in my head once the top right corner and plus Y K +plus said so we should do is choose case a B minus the injured upon the backs L to minor seems pulse of wine and then on to the minus the integer parts of hazards plus KY which is said minus 1 acts so that tells me what not come that tells me what about pirates not pie from G to this fundamental domain it is given by Paul A. All for 1 and 1 1 6 Wise is equal to 2 1 1 1 black expects brackets why fractional part and then brackets presented minus 1 into politics I believe so that's reductions of fundamental domain 9 take any snooze function we like on the fundamentals of mine and again I'm going to ignore edge effects take after all 4 1 1 1 X Y Z To be each of the 2 pilots said so it's news on the interior of the fundamental demand that there's some issues about what
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happens at the edge and then we get and automorphic function a gamma automorphic function fly from GCC given by find all of this matrix 1 1 1 X 1 Z is equal to each the 2 Pioline Z P minus Y brackets X is quantified exercise to verify gamma and variants of this just by hand and the problem is it's not quite as smooth functioning but let me just ignore them it's not quite legitimate religious has finally is in place is discontinued it's mildly the not bad discontinuity is that they are discontinuity so that's and that's what an automotive function kind of looks like and then to get a no sequence Michael just apply that automorphic function to a polynomial so what would be some examples of no sequence so all all from minus Peter and brackets Alpharetta and on each other gamma and squared minus Peter and our friends are examples of no sequences so this very instructive because you can see that we actually allseeing some higherorder behavior here so there's a quadratic phase behavior here but the key point is that there's also sought a more general behavior just slightly more general behavior involving what we call bracket quadratic so that's important in the theory take questions on the competition apart from How would continue to write off the list I'm gonna stop quite soon I think just on physical grant you what did it that's going away so many others riders on this board is a key point we get quadratic faces and slightly more general but brackets faces so you might think I I I used to think that why would once again in this summer some slightly abstract whereby no and groups in last season also moving function when you could just right this time this is a perfectly concrete thing constituting opposite point of view which is that if you ever see anything like this you should interpret it as coming from nocturne group otherwise you get into all sorts of trouble sigh much examples of the kind of trouble that can arise on a little bit later on visit to finish I want to Justin quickly introduce something called the inverse conjecture for the GAO known and inverse conjecture for the guards norms is In a statement of the form my it's the statement that says these are the high road characters so in other words given any any kind of linear problem like finding Fortum progressions it's an offer to work with these generalizations of the additive character as a sort of sufficiency therapy I'm 1 minds in this lecture is to just stay to the things properly or not this lecture 30 surprisingly difficult things have really state property and added that ever done said in a lecture anyone hangs around tomorrow and I think anybody's ever stated it problem election below a world 1st as these things go the GAO norms I'm certainly not going to have to prove it in this course of lectures and nightmarishly difficult so there was no arms or family of norms on functions which control essentially all in their expression said there is
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a family of norms on which we call the carrots you tune on the carrier's you 3 known and then for long and hard on functions ash from the interval wannabes went to the complexes and this is another receiving radiation identified mentioned this yet this is the same thing as interval 1 2 and those values norms and you can measure the size of a function with respect to them and they control quite general their expressions on the way in which they do that is why things statements known as generalized von noninterference in head so you all too often to generalize from 1 of the rooms in it such as I am his 1 60 solve some all that 1 would have to have 3 is bounded by a constant times the guarantee you to of any 1 of the FIS I went east of some is counting solutions the Expos White was that in those functions so it's 1 of and squared times summer of X and Y have F 1 next to 1 F 3 X plus 1 foreigners you before use where should be look outside let's just as it restricts to or they could but let's just say that I freeze it's only defined for up to it and then on another such statement would be Tcells fought for a piece of that 1 and 2 3 and 4 time is planted by the GAO G3 loan these people so you know this is an enormous home and homogenizing here Sergeant missed an assumption these advisers space to be so if 5 X bounded by 1 ways and so is a different such statement where here naturally T sub for it he about 1 has to have 3 and a 4th um is equal to the average verdicts in the office of 1 of 6 times a day for the next plus 3 again if the adviser banded what country they statements for the simple reason that I haven't told you what goes norms on so there would have to be a prerequisite at the end so here what I call tell you what the 1st 2000 off and you can now you could probably extrapolate the definition I'm going to be going over this again later anyway so this is the average the overall X 1 H 2 after the accident after the exploits H
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1 after the explosives to further explore 6 1 plus H to all of that to the power 1 onequarter I the average is a lot will choices of X H 1 and H 2 for which all 4 of these points lie between 1 and then as a symbolic come back to this latest away right now and then the EU3 enormous same saying that sort of generalizing obvious ways 2 3 dimensions pageone 1 page to page 3 that affects the threedimensional also the town 1 of 8 and then the sequence continues in pretty natural way but is not obvious that the norms and again I'm a sketch the provide later these generalize from nominations are actually not very deep so there and she just applications of the kosher shots inequality although slightly painful 1 so these followed from Monday because she shot I'm probably will show you 1 example of that later on just so you believe me I would want to do in general but only because it's notationally nasty and not because it's conceptually difficult in any way so once you have this definition of a is normal and this nation of generalisable Norman his strategy for countering arithmetical aggressions and set Spanish strategy for counting let's say and and professions in any say here is just some said given to United said Prime's might be some of the Sept how we can account for digressions in global with 164 A P 1 8 1 1 and that 1 or its equivalent about and that it would be silly to just try and all we can to try and pledge their straight away because we might not be trying to show that this squash the small so what you do is you you guess main terms in some way to Split 1 a as a structured part and structured class what we call a pseudorandom part at random it a in a suitable way so is a way of doing that for example that specific to the prime numbers which our nation later on and and then TO SEE for AP of 1 9 1 1 0 1 0 spitzer some of the structured part and structure 4 times plus 15 other terms each of which is around park 15 random nothing the ending is said Causey linear forms you know with 16 terms will expand such as tea for a piece of round up to half so with lock you can just evaluate this this is supposed to be a maintenance and then you could try ensured that the other 15 terms all evidence and small and then it's sensible to try use this ceremony here so we need to do is show that the garrulous EU3 normal after is more desirable to show that the towers that round Karadzic's reknowned in small so that motivates the question of how you show that how would you show that the guy was normal but function is small and that's a question this talk about mainly next time so it turns out that is intimately linked with the notion of no sequences and in fact the inverse conjecture for the cows states that a function as small Gallas North Garrison 3 norms said if and only if it's orthogonal too 2 steps class to know sequences so next time I will but I will be talking about progressions much now but I will introduce Gallas known to gain and remind you what no sequences are at much greater length about and the relation between the 2 but I'm totally exhausted from and you want to so I suggest that we and go watched him Gallas talk about time logic if you want few award