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1/2 The orbital circle method and applications, toral eigenfuctions and their nodal sets

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Automatisierte Medienanalyse

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Erkannte Entitäten
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a total of 2 and and not to when the sanctity of the organizers for the invitation so today just because there's no question what language they should go to to in English no good the topic and then going to discuss this it's mostly joint work with the the Addicks component and there there's is also some ongoing work for the general the general team that is the city that the general came is through understands that the sets of indigenous which are not exactly you what would you make call stand up to the attic medical objects objects but what used to be in the overly known hopes the ball saying the all bits of subgroups of Sept groups opening of renewed hopes and these are structures which can be purely comical but purely ,comma this so that this is a team that has been the study to for while they're in in different forms a mainly in the initial impetus input the was emerging offered a mutually about spectral gaps which a lot to do same saving a Endesa in his cage but then this is the story has involved and no 1 but 1 dishes and really wanted to understand what introduce you get in a given all so what but what they will do is given all of you the offer of some of these results they think it is better to give an overview wouldn't think to give due to go into into more details because the details are quite lengthy and that basically I wouldn't be able to say too much and also probably would be lost quite fast so because it had to to to do something which is not a technical just want to focus on the the main ideas and the methods by which are involved the sewer what they would do is give you a little bit of a introductory material and the back onto the on the level of the techniques which are involved in the thick of the results on the spectral gaps which have been involved in the in studio and also that accounting methods and then what they want to discuss our certain applications and unnatural applications and will see that their applications that do what this goal to in Gallup alone certain beckons also the 2 the problems along with the Nationals the whipped the ball oppression cautions and so endless conjecture and then we have also ongoing work with the conclave which which the use of onto the Duke Sugiki distribution Cuellar in the quadratic fields and the problem of range of this it's so the 1st the 1st 2 topics will be basically in the context we would fit the context of this local to global problems and the last 1 will be somehow what would could look like an natural application of this problem of sitting in in August so
let me give you a small in mind of what they want to do is just a mantra since this is supposed to be a little bit introductory so undermined you what is the holder committed works and why somehow in the present say what the elements that and only in the context of this group of cynical all bids to the fallen to have a chance said of performing in a successful way certain that it so basically what we have what we would look at the set of individuals which is produced whiled away a which is a substance say off of 1 or 2 and the the way we're 2nd Circuit say we haven't a distribution on 1 of 2 men which I will normalize for simplicity to be probability distribution and what we I'm expecting is that it is Laferney icky distributed and then would like to say that basically the support of this distribution is almost everybody so do give you a typical application related to the tool but there said a letter to the gold but problems and the badly golden goal but Poland you would look at all individuals which can be it's a obtained as the average of 2 old crimes Bombay can and will 1 like to say that to basically get everybody in that but 1 can't quit that but you can with what is going to happen is that 1 can show peace with the president acknowledged that the exceptions such as very small so this is this is this is dead so what 1 have is that you look at the forward transform so we taking land that would take its food comes from and we're getting some kind of exponential some which is a function of the circle and then basically 1 studies now under told this to US financial center and yet is the following it's still make a subdivision of this so the NGOs will advance formulae capture London-based s so the idea is that you make a subdivision imagine life in a major minor knocks by restricting this and they go 1st to the neighborhoods small neighborhoods for nationals with a small denominators so there is a gap of bombing to them which is being and which is typically going to depend on the so you're looking at pressures on the form payable through with jurors and the and the neighborhoods which oversized you will and so it should look at that as a tiny said that some of the views of the city is going to make contributions than there is
so no what is this what is a single 12 well what you do is that when you go back to the lumber which is given as interest would transform of this and district is integral to to the set of major so then there are several ideas 1st of all what you want to say Is that so this is a new 1 is a functional under 1 of the and Islam no 1 of any will tend to be better understood so the 1st thing and this is what they call the complementary products is that you want to say that London and the number 1 and a half kind of close together so the user way to handle it is better looking at but 1 way to handle it is they've been looking at the pool norm of the difference of London and the 1 which is of course by bypass about controlled by the pool in all of this of on the complement of the major arts so remember the plan no 1 was normalized so it would integrate and all of a sudden would have won over there so the idea is that if you if you take all these these major parks and is of course not always true but in the end in situations where we can play this game this is what happens to get something which is small with respect to 111 so they what it means is that whenever you get in the support of London 1 a this is not going to differ too much from what you get in the support of London and the difference is in Algeria which is going to be the over the world domes of size of is going to be a bit of the size and at some so depending how Spalding to pick up some this is going to add some that assignment send it all day and be the only if no I don't exclude that the say even under 1 is not known to have the full support of the also said that the set out 1 not necessarily being all of 1 of 2 men but at least there was going to happen there that's the idea of a stick into these major arts is that we have
some kind of better understanding of what happens they sold is London 1 and which is an intergovernmental Small said just near-neighbours neighborhoods small neighborhood Office of small nationals a well that what is typically going on is that the the number 1 and the scandal going to factor as a product of so you will have enough comedian density which is Sigma end a story which is Espy and which is just the yeah In that case would be of the order of 1 and then somehow there is a factorization which comes from factor this these modern IQ that leads to public presentation basically presentation in a product of over local densities and so the reason why we can't do that is that we want in some sense to have a very good understanding of of say the suspension of the initial distribution and a from the same model of point of view when we're taking the mother sufficiently small so when we went when we're looking basically we have to understand what this and the number of the set of Inter just less than an area which has a prescribed congruence conditions said to be equal to 2 a pupil rated Q was not without its once we have these understanding we have haven't good too good too we get a good description of what happens on major so the magnetic argued that there is apparently to be involved and so on 1 hand this understanding here is going to be easy win be small because we're taking very small neighborhoods of Russians small small-denomination on the other hand the base to small now and the end of the epsilon which were getting there the shortest
this epsilon is going to be the thing is going to be larger so the whole saying it's kind of a compromise between House Small power being how logic and did to get a small adjustment as possible and how well how
Logic Inc the believed to have still the control on the major
parks so now and this is a story that has been repeated over and over again but let me tell you why why in the context of the orbits of groups of Sunni groups we have a chance to carry such a signal so 1st of all the major acts so like it's clear from what they just said that basically what we need to understand there is the distribution of the set of former the model of fact and that Canadian model of point of view but only in a quite modest way in the sense that we need to know something about the distribution of elements molecule with you is not too much and it turns out that there actually this is possible because of the Spectranetics still discounting can be performed the by basically the last Phillips so 8 of the last exterior although all the negatives which I'm going to give you a precise accounting errors in the orbits of this of this all of these groups and actually the so what underlined that the suspect gap phenomenon that they will call you and say initially what happened is that we realized that we could do that well of course you will have respectful technique also in common when subgroups of the given but what was realized at some point is that there that the suspect told gap can be made uniform in some sense although condiments and in some sense what would get what we got was an extension of Sobek student and so somehow although the the spectral gap which is involved in this uniform spectral gap is served this is quite essential in determining what really can be captured from from Spectranetics no in the early stages of the steely we were really doing everything only using the spectral gaps know some of the new development is that we realize that it was better than a tool to use only the Spectranetics on the major parks and then stopped looking for Samoa although techniques saying affected almost a mobile that's techniques that can handle the the minor acts the so in 1 it's easy if you don't undermine Iraq's so we're taking the nationalists those which have much larger denominator so we don't quite know so much anymore about the distribution for these modernized so we're looking at all the techniques and so what they're really powerful techniques telephone in the sense that they don't like well you to To have too much structure where 1 such technique was developed Babineaux god of which of which there are estimates on exponential sums which happens to be structure so that basically doesn't require you to have any kind of enigmatic importance and we really need to know that little so why can't we get the such a new structure where we have a cynical so we can move multiply cynical clavicle promising legal so we can make it elements and because they committed by elements somehow you could expect that there is a hope to get there and the killing of structures being certain other instances of what can be used to although like a set this this goal is not to what what my colleagues peak of Sirnak likes to call it a single which means that it's it's not the it's not the Finance Index example before nicer algebraic Gopen since there so although this caucus ,comma Nicole it may have some nice subgroups and we can exploit his nice subgroups to make estimates on the Main know what they want to to below is style by recalling a little bit of diesel spectral methods a basically yeah there are about 2 techniques so that the delisting the offered techniques a luxe Phillips and then there is the symbolic techniques so it will say you
something about the the 3rd when they would give you the little bit of the
back chomp and then details some details along the market methods and I will not say too much about the the symbolic techniques which are all of which are more complicated the the idea
is following essentially like a city you want to do accounting eventually you have to say so basically the samples they should have said that it was on among the on the slide but anyway so that the book so they either sitting in a symptoms your isn't true the over goes into just so we have to do some kind of counting in all bids so basically the easiest thing is to have generating functions which are those which are constructed using the group amends himself and then we will basically condom for not million the full anatomy did not just a the the other thing is that there is this it the not that I look it but the question is is it taking the such such as several phone number central 0 0 this is tool that goes into just This is a story follow followed suit to work itself toward the view was you situation simply then you have to replace age tool by HP sold as a complete analogous to in In the next thing which is what would be involved in the study subgroups of the overseas to work overloaded all the goes into just as it is going to be the case in for his opponent but things anyway basically did this observation that reduces you do the issue of reduces the issue to to doing some kind of hyperbolic counting wail away on looking fine due to know exactly With an asymptotic former What is the number of of Entenmann's GE saying that given the temple which lie within the ball which is defined with these new here over so whatever key and then we can further also involves where we can impose certain brought as conditions but imposing this conference conditions amounts to respecting the element G To the Congo subgroups of the given the given subgroups now what is the interest Over looking at it
from that point of view where there is a killer but which tells you how to handle them so let me remind you that the background so what we're going to do is take a fine generated subgroup or listen to evil as to what goes into jails In what we assume is that it this so moment mentally in the sense that there will the minute set so you have a limit set for this goal that it has been mentioned so In this to case so it will have a dimension which is what we assume that we have a dimension which is the District People's so it's going to be typically less than 1 because we're looking at a single if you out India case than the 1 has to be so it's a great place there and blocking your view of course I don't know what what side of this machinery should stand on the list that I can change so this is what you get so it meaning that case officer to be the Delta is going to be between 0 and 2 Saudis and a critical level there which is also in the 2 cases a hollow in the plea cases Wong so what happens is that developers his mother you have know to spectral healing and so you can use this relaxed technique to do accounting but which gives you an asymptotic fall now will the leading it's basically given by the date that the law was taken value as we see moments soldiers of more analysis methods for the last 3 particularly basically reduces admission to having a spectral gap sentence which is then set when Delta is 1 powerful another 1 how you don't have enough to spectral cheering also sings There's going to be a little bit more difficult but you have know I still have indeed so-called Nantucket the symbolic approach in solving the symbolic approach we are not talking about spectral gaps over the Laplacian but we're talking about the coffee regions Elizalde In basically the well this is a cheerleader has been developed over the years and the new input is that somehow in this chilly weekend incorporates an additional model of point of view so we're taking would not sit too much about that what we succeed in doing it's extending the the main object which governs system of the medical report which is the Lovell comes from political kind of fine extension of his Lovell Council played into the model of setting where we can capture the modest thing in a uniform way and then we're getting worse uniform control of of the of the seal of the legion of these results of displacement in much the for what he's done with the speckled gap and an extra exterior so this is more complicated that we're not going to match in that uh that in the the deadline certainty that vantage of the 2nd method is that we don't really need the food group so it's it's small local and practical this 2nd technique is the moment of pigmeat applies also understand sitting wanted them on how to apply the 1st kind of techniques when you have a single so they was talking about the design of influences that would typically have a cynical setting where are we we don't really have old although the cynical quite sickens since we don't really have existed the most men so let me
remind you what his elects Phillips setting so we have what happens is that we can look at the Lehman selfless which is obtained by a court in age by number and then with taking the functions best which are invariant under the action of goodwill plunder and we're looking at at the end of the year the spectrum of that like said the you think in that case we can exploit and to spectral purely because well the most arrogant value numbers which is dealt 1 9 dealt with built is mention of the minutes said there is going to be separated from the rest of the of the spectrum so in the beginning you have you getting here .period spectrum and then a ball you're going to have continuous spectrum but what's important is that between 1 0 0 and number-one 1 this get Norway's is important well there is enough the is is a very lax and Philips which is kind of exploiting the the 2 sides of the picture is CPD point of view which stones which also on the way integration and there a genetic point of view and a word that allows you to do use expressing accounting so little was talking about his hyperbolic counting so we fix some element that we use your age and then in effect in is the Museo and W should be should be I say and then you're looking at counting all the elements gamma in London With the property that the hyperbolic distance from Denmark debuts yield to tool to W is bound by its so that would in some sense that give you a elements gamma for which that before being his nominees by exponent of this right so the point is that if you look at his former you're getting some terms here which comes from the the from the discrete spectrum there is an unacceptable here but even in this form sound it did leading the leading contribution really cans from the Delta's so that there is a relation between London you find out G babies fold up and so you see that and the main contribution is going to get from these numbers evil assuming that there is a certain gap between London's you number so if we can establish that the you have but is gap then basically you get a very precise the evaluation which is that the contribution of the of the most mold which would give you end about to Delta that an announcement comes a day and then something which is not only small about going to have some power gain because the the Delta 1 is going to be really separated from the delicacy at and we would like to have this phenomenon as uniform as possible and practical we need to have this phenomenon so we have this phenomenon in the in the given group blunder but will also have to understand this phenomenon in the in the congruence subgroups offer of Islam that because we need to to do the counting not only for the 2nd meaning a known that I would also when we imposed restrictions on on Gannett which out often of a model of that gang so what we also want to to be able to to count out there is a member of the garments which are in on the Q well unlike you out the elements of London which I didn't and which have is Bombay so of course for each individual Q we still have this this this format but then we want to know how this gap is going to evolve with you and this is
exactly at the point of extending so that's still to this the 2 descendants so good in the sense that you're no longer just all of us have to and then well how does this gap behaves well it's his uniform of Q and whatsoever but proved is that it's please the people of 16 in the infected conjectures that it should be at least equal to 1 quarter which means that in the picture was giving you before you don't have on and see you
don't have the
accelerated don't have this exceptional aid and that was that so what is the 1
involved in the input is the development of the celebrated the killing an analog of soda forcing groups innocent with so we have the same said that we have these long enough and could generate that none of the men pleaded innocent the and we're taking the confluence of groups and on the q of London if we cautioned a trouble on that Kewell we're getting a big month service because they are more functions which are going to be invaluable actually run the you so because they get more French there is a chance of spawning spectacle gaps well there is 1 so 1st of all what they will is that again dealt lumber is bigger than the health the ways this story falls apart and so isn't easy observation is that the boss told remains the same the problem is what happens with the
a it see the problem is
what happens the weekend indeed next 1 so what we proved is that the next 1 London 1 overrun the Q is going to be separated from London as usual by and that's which is independent of Q and is going to depend of course possibly on long enough that in principle the steps is effective but it is quite small so at the time well we were planning to to play in fact when when the developer is getting close to a one-half of their listeners an explicit the year Alex gamble which allows you to to make precise statements but still the somehow the use of the spectral gaps they the type of of singing and I'm talking about said would talk about the level of distribution in some sense it's quite restricted area In any case the presence all such the existence of such a spectacle get is still quite enough to deal with the major acts in the applications and went to discuss and so will definitely losing but that results in a quiet kosher way 1 well I want government about the pool here we are talking about the John expect will get no easier take guilt to space on edge model the but with a little bit of of abuse of language in incensed sense you can view it as the helped to small loan wait to send vector valued extension and his victim about extension it's after the the the Hilbert space for listened to queue intent of planned to summarize the the pool of of result is joint to the German paper with example and people sonic assault that what turns out is that if something goes along with the spectral gap it clearly has to happen in this extension was to it isn't to Q which eventually amounts to saying that something is going to go along With the spectral gap in the K-League enough on its own to Q which is induced by so this argument which is not a lot to completely reality there would have to explain it would take an hour but the point is that we can reduce the problem of DeJohnette expect will get to the common at all spectral yet Endesa Gailey which are known the very well understood so what we know is that so we do induction motor fuel we have been in the year the strong approximation principle that would give you something which is only 2 1 then we have the J-League gasses which have induced by that this and what we have of so-called expand families and we get this uniform spectral gaps uniform in the sense that the uniform of Q that eventually translate into his uniform uniform John might expect to get was is coming here so the consequences
than from what I said before is that we did not only good comedian company but also imposing among other sections we demeaning terms and an analog don't know what's important about the scale of the visitors with a ball game so that it's on the spectral get there is an effective cure of course but because of the national game here can take your questions and you can becuase lunch and which is kind of nice and well because of this issue about the uniform books and the strong approximation of we have to to assume that you will school plan would be given fixed modest users don't know if will not plan which was you know there's another there's another former possible that also gives you a description
no before I go to that
and let me tell you that in the other case so know what happens when window is a house owners and house where you don't have that anymore but you still can't produce a former for the for the comedian counting a using this term on the medical techniques at the end of Council operator which was at that time it was just to maintain them and then something which is a little bit main down but then these things have been developed felt that this is an important input by the people and the Peotone and no and so on that clearly have done this this picture this technique in something which is very quantitative game which would following gains and then what we had to do was to find some kind of extension of use of this technique will you also involve a model of a distinction here so we had to modify this well Council operate collected at the bigger spaces and then play to pull that you still have this as usual the region's folders all and then in fact will not Homeplus unsuccessful bid in the sense that we did not quite have the following gains but still we had something which was strong which is strong enough to b almost as good as this problem gaining magnification whether this will affect some of the votes that were mention that let alone so in some sense whatever games so there is an extra problem here with a smaller restrictions so what we didn't really do is to extended over particularly in the model of a modern aspects so all the information form this model aspect translated from Groupe expansion and it is something which is slightly weaker there's is another little problem with With which is different with the symbolic approach so like I said if you have about here which is close to 1 then have gambled which gives an explicit indeed symbolic technique you while you're going to get the sounds some asymptotic like that but would level again which is going to be quite good is going to depend on the media on initial set of generators so what you may always states say assemble cynical that looks much bigger so some of in the context of his I'm a problem with taking a cynical which is much sicker and we expect that you're going to have a better a small area after some sense but we don't know that because it's not really clear it would be horrendous to try to figure these old it's how the game and how the airlock demonstrated depending on their own the initial generated so this is another difficult so the
1st application there just to illustrate this feeling is that the problem of understanding encircled by kings in fact it is administration of the corresponding Keeley in there In age to but like a set distillates completely analogous and the spectral the problem there is going to capture the captive by this lacks Phillips those techniques and applied the cautions of Peachtree so for those who are not familiar with it what we are studying a cover just offer of surplus which have obtained the in so called upon on encircled by kings and the main point which is something that was itself based solely at the time is that if we're looking at the at the circuit backing will the initial Teresa here you would have the minus 11 is because this is because you're going into the interior also have that the circle than I have but 1 that 1 and that 1 so we have this we tangent circles that the Ali 11 24 21 and 28 into just what happens that this is not so difficult to see any witnesses in nice observation winger facility is that when you continue filling in the holes with tangents was using a belongs the Apollo still and so we have here the circus a 24 21 and minus 11 showed up at the looking at the similar to the dungeon circus you get to 28 and then you have a compliment here which is faulty and then you can keep going on and that the filling all the holes will all these circles are going to have the Indigo Girls just in the question is what into decidedly getting so
the volume was studied
in a sequence of PayPal's by John medallions models model Wilson down and then it was also the 1st various people some market values locations the this is in fact in the beginning that we will just applying this spectral gap results in problems of of seating so we wanted to know whether we got crimes in another In his beaten and things like that but somehow the the city conjectures which were put forward initially it was the 1st conjecture which is a modest conjecture is that the collections in any detect take any up alone and in the GOP opponents of protection and you looking at the corresponding coaches well the collection of all the skeletons they form a set of integers which have positive then b mobile to know and not a sink it is formulated also Inc in the work of these games but it was definitely definitely formulated based on my plate on his most likely this thing is true is that you have a local to global principle in the sense that old into just about reduced that to symphonic Conlans condition so I will tell you what this what is known about that for the 1st problem has been solved the 1st problem is much easier the 2nd problem is not solved and it's a little bit on the level of what they set about gold but is that we don't really know that you can get any integer some of the plants that even integer but still we know it's true except for a small exceptions said solicited bids for the the
same kind of situation so what has this to do with the proceeding and we all bits of groups well 1st visible on hope that is a subgroup of your total group corresponding to a quadratic formal signature of the 1 which is the 1 and I think they're so in other words what this the vanishing off this quadratic formula the vanishing of this quadratic form is a necessary and sufficient condition for all the circus so if we have a configuration of the circus with traders it's 1 expects Leakesville which almost mutually tangents and then we have the vanishing of of this quadratic form and so basically the way this up alone in this opponent backings work is that we have the initial Fokker also of the Basic Law of of the of the foreign given much attention circus and then we're getting the other good knowledge of the only time buy by taking the actions of elements in the subgroup in the office of stolen gold which is called the polonium into group the action of a on on what we would call him world portable so we are acting on this faltered here they can enter any amount of a acting on this fall to below and we're getting another 4 article which also covered just of circles In the backing associated to these initial circles and they doing that we're getting everybody know what is nice about this up alone in backing the hope is that it is found generated so you have full elements here is 1 of excuse for which are generating his opponent by Campbell in so now OK this is what has this got to do with it so too will goes into just well handled the sort of 2 features of group which are important in taking them if you see what happens in the spin devil of a you of it leaving his goal is the image of the subgroup offical to two-wheeler goes into jails which is frantically generated and which is nice in the sense that it's the dimensions of the minutes said it's going to be bigger than what the men believe we are in the age of the Internet in that situation the 1 house has to be replaced 1 so avoid showed is that in fact you have a group of dimensions exactly executives that so it's more than 1 that corresponds to the house of the mention of delays the also dimension over the residue of the residual area in the In a given backing so Spectranetics supply which is always good but still below the goalposts it's not very sick and because of Far Far Away From Home to so in what couldn't possible to use gambits visible that's 1 feature Bennett another feature although it is Gopi certainly a single opened to there are some groups which have very nice because there were times the group's generated by this particular elements so take
this 1 is this please any 3 of them in terms of what that these groups out of magic and they're all bits that can be described by banner quadratic farms not if you have to pull sings about sets of integers which can of produced by added medical objects like a barnacle that because you haven't quit coaching and the show In what I'm talking about there the 2 was fixed to the spectral aspect and the business the presence of having small enigmatic subgroups which not going to be important so what elderly
sells well administered following results obtained in the pieces of an inner folks they say sun approximation properties we know that if we do reduction molecule we're going to take you were going to get everybody provided due this week than that supplemented job so what we would have had to be understood there is worked out precisely this that these exceptional places so it turns out that the which proved is that the only problem scanned at at 2 and at 2 and 3 so in some sense of what we now in terms of local obstruction what what will get in there and there's there's this what about the possible local obstruction so the appeared at to end at 2 so what we showed using mainly due to the subgroups which would use this is but not a quadratic farms and you also representation of infidels Bay families of of binary quadratic follows what reporters are disposed to desert conjecture holds this is a joint paper with folks and then later would come dollar which will use some of these techniques that combine also with the spectral duly to pull mops and show that we have urged the local to global principle a in the sense that we getting everybody which is at admitted From the local point of view with the possible exception set which is small in the sense that we get the ball game here of course it would very much like not to have this problem but the difficulty some help to explain the difficulty it is basically the same level as trying to pull well that that's what we feel that maybe we are longer than are of them something smart to do development of basic on the level of going back on legal but with so the why and how House
is this input into what the well what will doing this time so mammal the circle meant that there was talk about creating a generating function and they want to hear the generating function so that they can as well exploit the spectacle clearly s as well as the additional structure well in that case involved to plead the mind us what they want to be able to exploit this enigmatic subgroups so I can doing here is the fact that he's so we want to win looking at individual so basically elements which Obama ability and will affect is a product of 2 0 and it's square for some reason the CEO is going to be very small small power of the mainly speak Square which is the main thing so why do I think square where he and when to introduce special elements in my alone in group which helped from 1 of these enigmatic subgroups in the state the following information the way they appear what what only matters he is is that it is the boredom all within getting here quadratic forms in Iksan why then so we are putting here where multiplying so we used to the groups that appear so well that Japan with the gamma which is in a bowl of latest easier In a four-ball so here we can exploit the spectral spectral expects it will take until 0 small it doesn't matter if it's a small part of the news is going to to survive so we let this operate on the quotable so utterly getting here are quite articles all the coverage of that will appear in the corresponding into government Sokol pecking and speaking Old the phone Collins so what we producing here archaeologists inspecting and that's what it comes out is that this is a good combination know why is this a good combination
well on 1 hand we having big seeks Y which is completely explicitly are quadratic forms and it's why and we can use that to estimate the minor acts note that 1 quadratic for 1 quadratic form wouldn't quite do it because you only have to wells but because it is also the Galibier we really have a fan of quadratic farms and this is enough to get the control which is needed on the mend on the other hand we also have to control exactly what happens on the major and this is why we have the gunmen here so the Gunners restricted to football .period cheer have this precise Luxford stomping and extension to the medical settings so that the control of what exactly happens on the major out scams from having the presence of of this job and then well 1 1 play complaint with that and eventually 1 gets policy dating back to the whatever estimates we have our very very far from being able to pull that we would get everybody so we can get to know of them which is just small enough to sense when the pollen in the it's more this winter with stops what we have benefited from so this is 1
application the the other applications so this was an application of the some of the the cheerleading the Indigo accepting what they want to know this you next is an application in the semigroup setting which is all the natural gas which is due 2 7 busses conjecture in the world this has to do with the understanding of professionals that have a bond apart from quotients so users problem that was
initially motivated by some of the multiplied considerations there what that I'm a conjecture is that even give you a modernists so let me go back 1 place if they give you
a In Ogilvy it is possible to find a B which is cooperating with the and such that defections be although has but proportions which of all of the capital with capital he conjecture could be taken equal to 5 and then other versions of this conjecture which will which are mobile Whitten acres 3 even on a cruise to provided but we take the sufficiently large so that toward the end of a finite number of all exceptions if this problem is so still open and there has been work they need lot of proving this conjecture for very special choices of the techniques that do not seem to extend in general so that the work would control which is in some sense a statement which scans of reasonably close to what to the approach that again we managed to pull off such results knowing that a small set of exceptions should maybe
stab telling you why it is that I am about to introduce this problem it has to do with distribution distributions of points the sewer basically if you're in 1 dimension and you want to distribute and point as well as you can see the island of the 6 ways to do it but if you haven't put immense is not so clear how you're going to have to put to choose this point basically what you want to do is a discrepancy which is the maximum of the difference between the value of there of over all of some satellite tangle with the number of points in May said that fall his like tangled not will not only taking will not only taking and squares with thinking general rectangles and in their mansion tool there is some so you can't do this to
too well infecting dimension to there is a
result of Schmidt which tells you that there is some indication of how well you can distribute points and the best describe what the optimal discrepancy is Logan over and so this is any choice was no matter how much was appointed his discrepancy is going to be bigger than so was shelters that you could realize
Logan by medical techniques by doing the following what you're doing is you taking the Sandy and there being which is called plan with the and such that the departure of portions of the ability of all then you are taking the following sequence in the unit square you're taking J. overloading and then you take BJ although will be when this is reduced quality so you get the points getting the points in the square and it turns out that this was a reasonably well distributed because the discrepancy is bonded by that same quantity which is probably all the 2 effective and this is factor is going to war to become in terms of aid which links scene they the significance of their Fontaine information about about fraction view this is also a problem and that's the deal in the context of the
Londoners because it will not looking at the special case of Dominican dimension to the random number generator which is obtained by multiplying would be what the and no translation so if would take the prime and we take the primitive elements not the then we can start looking so we're getting the sequence of the elements here we take over of the reduced not 1 studied distribution of all of the sequins and also studying the distant what scholars even more important is to study the source of the so-called distribution of the prayers of consecutive tournaments where the Coliseum correlation for but you mentioned that maybe this is primitive modern Monday then these immediate apology basically give me it wasn't all that is use Monday and so the discrepancy of that sequence is controlled just as the same as a discrepancy of the of the Sept speaks that I introduced before so
we want to what we are
interested in France is introducing a prime numbers the and for which we have an element of the which is primitive remote the and such that that departure partial portions of the overload you would be bounded by basins plays fixed
constant and these are pictures which will produced about Michael today have no idea how you could watch is going to have to put but you see that this is this is a it's it's really toward and what were designed by spending because here you have some instance of abuse of the will the profit cautions small that their metabolic bakery and when you look at this distribution Saiful to assume coordination and it's something that really looks like quite will distribute upon and if you
look at so this it's again here we have a prime number we have a primitive numerical but no all the the which has some large partial cautions and either way you see that these distributions is much less nice so this estimate of the number of newly flex the crew of the behavior that that he see saying on 1
picture so what are the results saying that it probably will stop soon we discussed and what with it so it can't this what we showed is that it will take a equals 50 DEA is a set of density 1 and even in the more quantitative form In particular the air lock down where they often can even be improved nobody's while local and this is because we are using the term agreement that this is just a shortcoming that can that because of the Of the 0 has announced the allegiance In the model of aspect which is not quite uniform but slowly deteriorate so this could affect there but there's there's chance when 1 can do better and maybe 1 can you policy there than in any case that result was improved fell below by a student of the edits control which dance the same who pulled the same knowing instead of acres 50 got a crossfire and in any case because we have a very small exceptions set here what is possible for instance if you look at it after 51 you would get infantry many factions view of the will of the spine and be the Senate it is this is a primitive ruled Monday so note the stated told the Dominion led to the conventional number generator is valued fall for that is that the spectacle of the people the while the what is
really well this is really fits in well a decision that all of the Of the goal opened
the same legal it's just how you put use such such creatures as you want to put used elements of ashes be over the week to Smallwood Bonapartian portions of what you the following make it clear that that the fraction of the Prussian portion of the overly is given by a 1 again this is the same as writing down and Inc and equality of all this kind of about but may persist they are not without becoming too dominant this Maine is 1 Whittaker don't be taking even you can turn it into what determines which I once all communications .period as a but basically what you have here is laughter speaking behaves like a cynical publicity and what we want to do is is extract we want to understand saying How many elements of the chicken producers as the lower right corner of such products sold products that they don't former group Donnie former cynical like to study that and the United States a large white we like to take a lot because that in some sense there is a condo part of this then mentioned built up over the goal and disgruntled about here is
just the dimensions of the set off the most between 0 and 1 that have continued fraction expansion Bonebrake Aso here were not talking about a rational numbers were going to the minutes worth taking the full set of real numbers between 0 and 1 with of cautions but kept the peso is a candle sets in so this campuses have been studied in fact also with the determined and committed practical about John Jenkinson Bodycote got to the precise dimensions of the scandal said that corresponds to a cruise to and what's important for us is a result of Hensley that tells me that when pays large this this dimension Delta area is going to get close to once in some sense that all of these Delta states that much like the Delta in the case of subgroups of visit so what we do know is that the reproducing the these they taking the title call and then the actions of Jedi In the world like to
follow the previous came and how we do that well but we are not going to take a football because then we don't have a neutralinos such what was going to take is a product of balls because of a central product of balls that allows me on the mind of access to do know God of note that in order to awaken always it is in a goddess and that in order to gain enough it's absolutely essential that built is very close to 1 and that even is scathing about the standard people that somehow is not completely clear of this really want to get enough have work this is in fact something we did before and the setting so it's kind of weird producing his argument that so this is generally don't don't the group's virtually just using the fact that we have vacation subsets of of balls and this was instances but we need is public expected and the number of the major house of course we are again exploiting the spectrum methods but now to be the deer symbolic approach because have a cynical so even if the cynical presenting sick we can't apply the Luxford it's because we don't have the demotic we don't have a good so we can't use is that the 1 thing that we can use speckled Wikinews this symbolic technique as was used before in the study of Campbell said in the information we are getting is sufficient but note that the but what you would expect is that when the NEA is getting larger because of Senegal is getting begin in some sense then we are better off on the level of the Major asked simply because we are expecting this these the OFI regional the resolve to get better and better and fortunately we don't know that because these hostilities messy and on obligated enough that it's absolutely unclear it would be a nightmare of Tigers down off House intervention dependent so on 1 hand we have to take a sufficiently large in all those who supplied the estimates for the minor parts but the effect is that somehow on the level of this she has a we could have something which is getting worse so this is a little catch and this is also going to be up there and in the next application will not I will stop here and this people desperate to him over the years they have to to stop you may be able to continue then the next time and they will distract you and another application of this this study of offer In fact it would be "quotation mark latitude actions the study of this ball apart from quotient the constructions in the context of old Duke steel and where in some sense showing the limitations of of Duke's can distribution Guillain-Barre but deleted sinkers should stop here on them and we will continue to believe that next time which the basically the last application which is going to be an application of that to work to language of physics and was born the extent to which few of the people united in at the Incirlik directions for use in my hand and he joined the "quotation mark use of the public that Wilson was sent away wouldn't really would include study it in detail but lately I think Alex told me once that that was you would have and plaintiffs Singh said please join the Globe is going to get worse I want to be able say that yeah really but yeah I have to give them the license plate clear but it doesn't really on my demonstrated that you too much in fact what we were hoping to find a quick to use this as the seminal case to the group case which would have made a lot of matches but then eventually gave up and so sensors anyway the young technique the symbolic thinking that works for cynical is just the player but there are certain drawbacks of the and this will be a problem in the next the question and and and
Nachbarschaft <Mathematik>
Distributionstheorie
Stellenring
Gewichtete Summe
Dichte <Physik>
Gewichtete Summe
t-Test
Zählen
Diskrepanz
Gerichteter Graph
Gruppendarstellung
Hausdorff-Dimension
Statistische Analyse
Translation <Mathematik>
Punkt
Analytische Fortsetzung
Auswahlaxiom
Distributionstheorie
Primzahl
Geodätische Linie
Biprodukt
Kongruenzuntergruppe
Dichte <Physik>
Entscheidungstheorie
Eigenwert
Physikalische Theorie
Ablöseblase
Ordnung <Mathematik>
Algebraische Struktur
Explosion <Stochastik>
Bestimmtheitsmaß
Archimedische Spirale
Dualitätstheorie
Subtraktion
Folge <Mathematik>
Diophantische Gleichung
Überschall
Multiplikation
Kreisfläche
Quadratische Gleichung
IRIS-T
Spektralmethode
Auflösung <Mathematik>
Sigma-Algebra
Mathematische Logik
Spezifisches Volumen
Gittermodell
Algebraische Struktur
Wärmeausdehnung
Kongruenzuntergruppe
Spieltheorie
Fächer <Mathematik>
Reelle Zahl
Hyperbolische Gruppe
Indexberechnung
Spezifisches Volumen
Diskrepanz
Automorphismus
Analysis
Gasströmung
Quadratische Gleichung
Zentralisator
Turnier <Mathematik>
Krümmung
Prozess <Physik>
Familie <Mathematik>
Gruppenkeim
Element <Mathematik>
Analysis
Resonanz
Untergruppe
Gruppentheorie
Existenzsatz
Eigentliche Abbildung
Uniforme Struktur
Analogieschluss
Parametersystem
Lineares Funktional
Äquivalenzklasse
Torus
Vorzeichen <Mathematik>
Topologie
Arithmetisches Mittel
Teilmenge
Druckverlauf
Orbit <Mathematik>
Garbentheorie
Beweistheorie
Koordinaten
Aggregatzustand
Halbgruppe
Lineare Abbildung
Total <Mathematik>
Gruppenoperation
Physikalismus
Nichtlinearer Operator
Term
Punktspektrum
Mittelwert
Stichprobenumfang
Schätzung
Zählen
Inverser Limes
Biprodukt
Gleichmäßige Konvergenz
Leistung <Physik>
Quotient
Relativitätstheorie
Orbit <Mathematik>
Physikalisches System
Objekt <Kategorie>
Quadratischer Raum
Flächeninhalt
Differenzkern
Rationale Zahl
Parametersystem
Innerer Punkt
Quotient
Momentenproblem
Extrempunkt
Desintegration <Mathematik>
Hausdorff-Raum
Gesetz <Physik>
Raum-Zeit
Computeranimation
Richtung
Deskriptive Statistik
Negative Zahl
Stetige Abbildung
Tangente <Mathematik>
Korrelationsfunktion
Addition
Multifunktion
Gruppe <Mathematik>
Kategorie <Mathematik>
Güte der Anpassung
Singularität <Mathematik>
Partielle Differentiation
Binärbaum
Reihe
Integral
Näherungsverfahren
Menge
Wurzel <Mathematik>
Rechter Winkel
Sortierte Logik
Multifunktion
Zahlenbereich
Konditionszahl
Familie <Mathematik>
Theorem
Teilmenge
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Stichprobenfehler
Überlagerung <Mathematik>
Spannweite <Stochastik>
Zufallszahlen
Rationale Zahl
Abstand
Geometrie
Bruchrechnung
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Erweiterung
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Menge
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Wärmeausdehnung
Hyperbolische Gruppe
Residuenkalkül
Grenzwertberechnung
Resultante
Korrelationsfunktion
Impuls
Abstimmung <Frequenz>
Punkt
Kartesische Koordinaten
BAYES
Übergang
Einheit <Mathematik>
Endliche Menge
Kombinatorik
Folge <Mathematik>
Nichtlinearer Operator
Zentrische Streckung
Bruchrechnung
Approximation
Exponent
Stellenring
Globale Optimierung
Expandierender Graph
Endlich erzeugte Gruppe
E-Funktion
Teilbarkeit
Rechenschieber
Matrizenring
Gruppenkeim
Funktion <Mathematik>
Ganze Zahl
TVD-Verfahren
Kategorie <Mathematik>
Punktspektrum
Gammafunktion
Bilineare Abbildung
Orthogonale Funktionen
Hausdorff-Dimension
Zahlenbereich
Kombinatorische Gruppentheorie
Unendlichkeit
Ganze Zahl
Zusammenhängender Graph
Gleichungssystem
Gammafunktion
Transferoperator
Schätzwert
Quadratischer Körper
Beobachtungsstudie
Kreisfläche
Matching <Graphentheorie>
Untergruppe
Transformation <Mathematik>
Logarithmus
Vektorraum
Ordnungsreduktion
Integral
Inverser Limes
Singularität <Mathematik>
Freie Gruppe
Quadratzahl
Modulform
Residuum
Mereologie
Normalvektor
Numerisches Modell

Metadaten

Formale Metadaten

Titel 1/2 The orbital circle method and applications, toral eigenfuctions and their nodal sets
Serientitel Summer school Analytic Number Theory
Anzahl der Teile 36
Autor Bourgain, Jean
Lizenz CC-Namensnennung 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/17016
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2014
Sprache Englisch

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Fachgebiet Mathematik

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