 Ok. So, this morning we have seen how we can use equidistribution theorems on the space of lettuces to prove something interesting for distribution of some number theoretic sequences of one dimensional sequences distributed modulo 1. And so, in this afternoon I would like to talk about more general problems of distribution and randomness in point sets. So, imagine you look at the stars on a nice night in Trieste, no clouds and you look up and you record the directions in which you see a star. So, the stars are points, you record the direction in which you see a star, which means you draw a point set on the unit sphere and then you would like to understand something about the distribution of those points on the unit sphere. Ok. And you can ask how random are they, etc. etc. So, that's exactly the question we're going to ask except that our stars will be sitting on particular sets, for example on a lattice or they could be completely random. So, we leave this general for now and ask some more general questions. So, just to make precise what I mean is here's our point set and this is where we are at the cross. So, we draw the unit sphere around this point set and now we draw a line from the star to the observer and record the direction of each of the stars as a point on in this case the unit circle and we now want to understand the distribution of these things on the unit circle and this of course works in higher dimensions. So, I'm going to denote the point set by P and by point set I will always mean a locally finite set in RD and we are going to look at all the points that are within distance t. So, obviously you can only see the stars that are not too far away. So, we define PT to be the intersection of our point set with a ball of radius t and then so this would be our ball of radius t here and then we are interested in the statistical properties of this as t tends to infinity. Examples of sets you might want to consider well ZD that's something or other Euclidean lattices or you can look at the primitive lattice points, you can look at shifted lattices, you can look at orbits of hyperbolic lattices. So, gamma is as before gamma is a subgroup in let's say SLDR that's another nice set to look at and E is just any fixed vector you start off with. P could be a quasi crystal and that's something we are going to look at. So, for instance you could look at the vertex set of a penrose tiling and ask how are the directions of this distributed. You could look at the saddle connections of translation surfaces and in particular this question of statistics of direction has been looked at by Atreia, Czajka and LeLivre or it could simply be a Poisson process or other random processes. So, these are some examples and a sort of very closely related question and this was the question that interested us most but I won't actually talk so much about this is if instead of looking at the directions you look at the free path lengths in the Lorentz gas. So, you could take your point set again, here's your point set in Rd, now around each point you draw a little ball of radius R and now you consider so this is our radius R so this will be 2R the diameter and now you consider a point particle moving around in this point set and that's the famous Lorentz gas and one question you can ask is what about the distribution of free path lengths in the Lorentz gas? This means that you start with a particle with a random position or even a fixed position let's fix the position but with a random velocity V and you ask if we is randomly distributed in some way with respect to some nice measure say the uniform measure on the sphere so you shoot with uniform probability in any direction when will you hit the first scatterer okay and the analog of letting T go to infinity here is the limit when R tends to 0 and in this case you can hope to find a limit distribution for the distribution of free path lengths and I'll come back to that so that's another problem that you can ask here and the techniques that I'll now develop in this lecture and with Andrea's will continue tomorrow will help you understand both the question of the statistics of directions and those points as well as the distribution of free path lengths as you will see okay so now the key message of this lecture will be very similar as in the two-dimensional setting this morning is to explain how these kind of the this kind of set of problem can be reduced to a problem of studying certain spherical averages on the appropriate spaces if our point set is a lattice the appropriate space will be the space of lattices if the point set is something else it will be something else and something more mysterious which we don't have a theory for we have just little examples that we study and we were already very happy to be able to do it for instance for the Penrose tiling so that's what we would like to understand this morning I showed you how to reduce the question to horospherical averages now here the the the task is to reduce things to spherical averages now in order to do this let me talk start talking about the question of the fine-scale statistics of directions of such a point set that's projected onto the unit sphere so again we have points the point set P and RD and we project all the points that are within distance t to the observer onto the unit sphere yeah that's the setting and now we would like to understand the fine-scale statistics so let me make a three-dimensional picture so this is the unit sphere on which we project our points this is the big ball of radius t which we consider our point sets and now do you remember this morning how we characterized randomness in a point set on a circle we simply said that we are going to throw a very small interval whose size is proportional to the mean spacing of our elements and we throw that random onto the unit interval right and we want to understand that statistic that's a good measure so here we're going to do the same thing we're going to take a very small disk or other test set here that's whose whose volume is proportional to one over the number of points that we have and we were just going to place this random on the unit sphere okay so we'll have a very small set here and it will now be equivalent rather than projecting and looking at the points here to simply ask what is the number of points in such a cone now the way we are going to scale the cone here is in a very particular way so I'm gonna take a ball of radius t so this is the radius t let me draw this in yellow and here's our little set dt the point sets that I will look at will have sort of like the letters more or less constant density so when I count the number of points in a big ball that number will be proportional to the volume of the ball so this means that my disk here should have a size that's inverse proportional to the volume of of that ball so omega will be the area element on the unit sphere here so we will now ask that our disk dt has volume sigma over theta d to the t and one thing the theta will be exactly the number of points asymptotic number of points so when we count number of points in here we would like to assume for now that it grows like some constant times t to the d okay when points grow at a different rate then you have to do different scalings as t tends to infinity so dt will be a spherical disk on the unit sphere with this volume so that's the right scaling to understand the fine scale statistics and let's just see how things scale here so when you blow this disk up to the to the to this disk here that is that the the disk on the sphere of radius t then the volume will be roughly or not roughly it will be equal to sigma over theta td I should have said here this sigma let me just remember what that is I think that's the volume of the unit ball or unit sphere what was that sigma I know okay so yeah that was that is just the maybe I do that sorry Andreas yes yes yes yes yes yes yes yes so okay yes I'm sorry sigma is just sigma is just a fixed constant relative to which I measure I measure the sphere okay so I just say I'm gonna you're gonna give me some sigma and then I'm gonna define my my sphere in this way so sigma is just a variable okay because that's my test set I don't just want to have one disk I want to have many disks okay so I want to have big disk and small disk but they're all gonna be very small relative to the sphere so sigma is just a parameter yeah you're right okay okay and now we want to understand the the fine scale statistics so what are we gonna do is we as just this morning except that we're now working on a sphere is we are going to look at the points in our point set PT restricted to the ball of radius T that fall into this disk okay and I'm introducing also the notation V here because the disk has center V okay so this direction here in which we go is V and what we like to do like in this morning we were throwing the center of the interval at random we will now throw the direction V at random so that's now our random variable so we're asking what's the probability of finding k points in a randomly placed disk of radius well of this volume sigma over theta T to the D and this is the same as the number of elements in our set P intersected with this particular cone here so this blue thing is the cone that we have here so this is C TV and how does this cone look like well this cone as you can see is very long because T tends to infinity it's a very long comb and up here it's very small so the disks that describes the cap the spherical discs that describes the cap of the cone is roughly has roughly volume 1 over T so it has radius here this radius is roughly what T to the minus 1 over D minus 1 that's the radius of this disk right it's a D but then D minus 1 dimensional disk it has volume 1 over T so it's radius is up to constant 1 over T to the D minus 1 so that's how it looks like and now we're going to do a very similar trick as before okay let me just formally write down what we're interested in and that's exactly the distribution of this number when V is chosen randomly so let lambda be a probability measure Borel probability measure on SD minus 1 on the unit sphere so that's the sphere of directions this means it's absolutely continuous with respect to the volume element on the sphere and we are interested now in the number of points in this little disk so this is simply that okay so this represents the probability that they are oh sorry that's equal to to R that are exactly our projected points of our points at P in a small disk in a certain direction yeah so this depends on sigma which we will keep fixed forever but we can vary it we can always decide which sigma to choose right and so if you remember this morning I talked about what happens if you have a very random set so for instance if P would be coming from a point so on point process then again as T tends to infinity this would converge to a Poisson distribution and now the question is well what will this converge to when we have a lattice when we have a quasi-crystal or when we have other interesting point sets okay that is the question is that clear yeah yeah this means lambda is absolutely continuous with respect to the volume element on the sphere right so that's the problem and now again I'll show you how to translate this problem into a problem of spherical averages and the key observation is that we can again act on our D with SLDR and convert this very long and thin cylinder into a nice object the price we pay is that then we will act with SL2 SLDR on this point set P so if P is a lattice we'll act in the space of lettuces and if P is not a lattice then it's not so clear what's happening okay so let us define a map K from the sphere to SOD that takes the vector V to a rotation in such a way that VKV gives you the standard unit vector and for the first coordinate so that's just that and you can always find a smooth map that does it for you that smooth except maybe at one point and secondly let me define a diagonal matrix that has in the first upper corner a T to the minus one and then down here a D minus one block where we have this so you see this matrix also has determinant one so it's an SLDR okay because here we have D minus one guys of this so when you take the determinant you get exactly one so why did I introduce those guys well the key fact is which you can check that when you apply this these two transformations to our long and stretched cylinder then this long stretch cylinder becomes and I'm saying cylinder I should say cone then this cone becomes much better proportioned so then so here is the guy that you see on the picture so what am I gonna do first well I apply this matrix and you see then what you get is a cylinder so I'm just rotating it so it points in the first coordinate direction and then if I apply DT so the cylinder now looks like this now if I apply DT what will happen is that the long direction will be pushed together and the direction orthogonal to it will be expanded and when T is very large you see that this will converge to a nicely proportioned cone that looks like this everybody believes that yeah do you all see this what happens to this to the cylinder as you stretch it down this thing has a little spherical cap but remember when this is very very big this is a very small spherical cap the radius of the spherical cap is of the order of 1 over T to the minus T minus 1 so it's very flat so if you push that together with this linear transformation you'll get a cylinder there that has now a flat cap and that's exactly this object here okay so what have we achieved is that this number that we are interested in X1 yes yes yes yes correct yes so so roughly and this is heuristic now roughly the number of points in the cone that I was interested in is now given by the number of points in this cone but for a set that has now been rotated and stretched let me write this underneath rotated and stretched and now intersected with the fixed cone now the reason why I only write approximately the only reason is because I have because this is a is a convergence so this is only roughly this set right so when you look at the difference between those two set that will be a very small area and whether or not there is a point or not will give you the error in this term what we will see later on is that the probability that there is a point in a very small set is very small so in a distributional sense you can make this rigorous okay so if you're unlucky these numbers could be completely different because remember the way we've scaled this set you have a very small set so we expect only finitely many points to to be in this set right it's a random set this cylinder if you like has a fixed volume as t tends to infinity so when I write this is approximately that it doesn't make much sense because we expect this to be a finite number and we expect this to be a finite number let's say one or two right but it will make sense later on in a probabilistic sense so that the probability by that these two numbers actually will differ will be very small yeah but I'm just trying to give you a sort of a general feeling for what we're doing here okay so this is the thing that we're interested in and now we are interested in the distribution of this random variable that's a random variable because v is random and you can think now of what you have here in fact as a random point process what is a random point process so what we need to do we need to study the random point set and let me give that a name XIT which is simply our original point set p rotated and then dilate it and what we like to prove is that this random point set converges in distribution as t tends to infinity to a limiting random point process and let me explain to you why that is why this problem is a special case of this formulation so so the question what we have here is does such a limiting process exists what size its properties etc so just to recall recall so a random point process you can think of as a random measure in RD and this random measure has a very particular form it basically assigns to each point in your set a delta mass at this point and that way you can think of it as a random measure and how can you test this random measure well you take a test set and you simply ask how many points of my random point process fall into this test set okay so what's the probability that for any given nice set you find k points of the random point process in there and you cannot just only take one set but you can take several sets and if you understand it for k test sets then you say you understand the distribution of your random point process in the you understand the k dimensional distribution of your random point process okay so a random point process is a random variable taking values in the space of locally finite subsets of RD and its k dimensional distribution is given by asking for the probability that the number of points of your process the number of points in test set a1 is equal to r1 and so on number of points in the test set ak is equal to rk and we are only interested in finite dimensional distribution here in fact we are only interested in the one dimensional distribution of this point set okay so my message here is if you take your test set a1 to be this cone then you answer the question that we got here if you're interested in the free path lengths of the Lorentz gas then it turns out the test set is not a cone but a cylinder nothing else changes so you see that thinking of this as a point process is quite useful because the notion of the point process doesn't really depend on the on the notion of the test set right it's independent of the test set of course it depends on the spaces of test set that you that you want to consider but in this way you combine both the distribution of statistics of directions and the distribution of free path lengths into the same framework and what we really want to prove now is under which circumstances we can understand that this randomly rotated point set converges to a limiting process and if we understand the limiting process then we understand the answer to the two questions of directions and the Lorentz gas okay so let's go to the first example where we can answer these questions and that is again our space of letters so P let's assume is now a fixed Euclidean letters and see what we can say here and you will see the parallels appearing of this morning this morning's lecture and the kind of thing that Andreas and I are now studying is in which cases can what we are doing in the space of lattices in which for which classes of point sets P can we extend these configurations yes yeah yeah so here these AIs will be from a certain class of test sets and in particular in what we will consider these will be bounded sets with boundary of measure 0 and then that's the kind of sets we are interested in and you can take so AI to be this cone and then you go exactly to this so cones are allowed test sets in other words but it always depends on what kind of process you you look at of course P so so in other words let's say P to be z to the D or general z to the D M for some M in SL DR or no no right we can do ASL DR let me write a little G little G and ASL DR so we can also shift our lattice and that's quite interesting because that means you observe your your letters from a point that's that sort of shifted you're not sitting on a lattice point as you look out but you're shifted against it was that a question the reference point is the origin that can happen yes so that's a very good question that's an excellent question so I have two options here if I have two points that lie exactly on the same line I have two two possibilities I only counted once or I counted twice and we can do both things and it's a very interesting question so either think of the set that is the projected set as a multi set okay then you count them button with multiplicity or you don't and then you just count each point once what is not equal this to that no no it is yeah okay so think of this as a multi set okay Andrea said swap it around up PT no no PT PT PT PT okay everybody happy yeah you're happy now right yeah very good okay right so so if this is the case then we've already seen what to do this morning so we now are faced with this problem of looking at the Lebesgue measure of V and SD minus one so that ZDM so I'm taking this on on G and then I'm acting with rotations KV and okay so I should now embed my rotations into how did we how did you define this Andreas did we define that okay let me not act like this so this is a point an RD so I can just rotate it for now and then I can apply this matrix TD so I'm looking at the set intersect it with a test set AJ and let me just do it for one test set a for now to keep notation simple and we are asking that the number of points in here is equal to K so you can do the same thing that I'm gonna explain to you now also for several test sets it's over there but it will just complicate the notation so this is the measure of directions that go into this and now just this as this morning we are in going to interpret this as a integral over S1 D minus one of a certain function so f is now a function on G mod gamma and I think you call it G prime mod gamma prime which is ASL DR and gamma prime is ASL DZ and we rotate it and we stretch it and what we are interested in is exactly this integral now what we discussed so far is what happens if this is a horror cycle now we have a spherical average and what one can show is that also these spherical average and not just horror spherical averages will become equidistributed I'm going to define f yeah but I just wanted to motivate it first so the question is now using our theory on homogeneous spaces can we understand that as t tends to infinity this will converge to a certain integral of f with some probability measure we don't know what it's going to be is it going to be harm measure is it going to be some other measure we don't know right but that's what we want to understand and as Andreas is already anxious to know is what is this function f in this particular situation and again just as in this morning in the one dimensional situation it will be some characteristic function so what is going to be f of G is going to be one if okay let me write G tilde so you don't confuse it with this G if the number of points ZD G intersect tilde intersect with our test set a is equal to so why did I now call this okay I think I always called it R over there right yeah R if it's equal to R and 0 if not so that's exactly like this morning right so though the idea is again you can show that this is a function on G mod gamma because if you replace G tilde by gamma G tilde with gamma in gamma little gamma in capital gamma this will be invariant so it is a function but as this morning this is not a continuous function and normally when you talk about equidistribution you want to show first that this holds for all bounded continuous functions and now we have a problem and this morning I didn't tell you how you overcome this problem maybe I did in words so the key point now is to understand how one can do this and so the key point is this is a characteristic function now if you can show that the characteristic function is the characteristic function of a set whose boundary has measure 0 then with respect to the limiting measure new that you have here then you can approximate that characteristic function from above and below by continuous function continuous functions to arbitrary precision and then you look at the limb soup of your continuous approximating function and your limb inf with sandwich the two and so the error that you make can be arbitrarily small and that's why also the limit for your characteristic function has to exist so what I would like to explain is how one can achieve that okay but before I do this and let me state let me so what I okay so question that we have is that when we look at the number of points so in order to prove that the boundary of this set here that we've defined here that's a subset of G mod gamma to prove that this hat has boundary of measure 0 what we need to establish is that the probability that there is a point in a small set is small and goes to 0 as the measure of this set tends to 0 so let me just see what the best way of saying this is just a second so let me just write this down a little bit better so we have this limiting measure new and what we would like to understand is that the measure points G D in a is greater or equal to R has boundary of measure 0 if a has boundary of measure 0 and the letter is with respect to the back measure I yeah yeah yeah sorry exactly so has boundary of mu measure 0 okay now if I can prove that then I can also apply this theme so in fact the trick is if you want to do approximation from above and below is to in fact look at the probability that is a set has R or more points in it and then of course taking my difference you also can take the set that there are points in it this gives you a better monotonicity because if you increase your test set R then the set that you have here also increases or decreases I always remember which I forget which way around it is okay while if you insist that it's equal to R then this doesn't hold so this is what we would like to show we would like to show that this has boundary measure 0 bound measure 0 boundary if a has live back a boundary of live back measure 0 now and how are we going to do this and I was a little bit silly here that I allowed ASL to R so I would like to go back to SL to R to just make things more concrete so if we are just an SL to R then this theorem then this convergence infects hold for all bounded continuous functions and with respect to harm measure as the limit measure so let's look at this case so we are now in the space of let this is as before okay so then this convergence for any bounded continuous function is in fact a well known theorem that follows from the mixing property of the diagonal flow and so now in this case we need to show that this has harm measure 0 and let me show you how this can be done so the measure of the boundary of this set can be estimated by the measure mu of an arbitrary a small set that sort of contains the boundary where the number of points between the interior and the exterior of this set differ so what you need to estimate is precisely let's call this set B where there is more or one point in this set B that describes the difference okay and what you would like to show is that if the measure of B is small this set will approximate our set the boundary of the set a from from the outside and the inside then or just from above sorry just from the outside then we are done so how do we do this so we can write this as the integral of the characteristic function of this set oh sorry there is the g tilde here intersects with B you want this to be greater equal to 1 and we integrate that over the mu g tilde and now you apply Chebyshev's inequality which simply says that this is the bounded above by the number of points in B and the great thing that happens now is that you have an exact formula for this quantity here that's given by Siegel's formula which tells you that this is nothing but the volume the Lebesgue volume of the set B okay so what you see then is that the measure of this set is bounded above by the volume of B and that's exactly what you wanted so if B is small that measure becomes small and that will show that if your set a has the back measure 0 then also the boundary of the set that we considered here will have mu's measure 0 so let me write down Siegel's formula and then Andreas will talk about generalization of this tomorrow so Siegel's original formula is exactly a statement about the measure mu on the space of lettuces and it says the following so for any function that's L1 on RD we have the following identity that the integral over how measure of this so-called Siegel transform which is defined simply as the sum over all in ZDG not counting the origin okay so you take your function F you form the sum this is called the Siegel transform of course point-wise this is only nicely defined if F say is continuous and has compact support but you can then extend this by the usual density argument to L1 and you get this formula and this is the Siegel volume formula which relates averages over the space of lettuces to the back averages what is equal to 1 yes if D is equal to 1 no no so if D is let me just because I'm already running out of time I'll answer in a second so the absolutely amazing thing that will Andreas will talk about tomorrow is that each extended this formula from lettuces to very general point sets namely to point processes or even random measures that are SLDR invariant and satisfy certain regularity conditions so Siegel proved this really by using number theoretic techniques and which realize that it's a very general probabilistic statement and we have then used this kind of Siegel Beach formula as a technical tool to also understand other point sets which will be in the remaining of this lecture so I'm sorry it was a little slow and technical but this is really a very beautiful and important formula and I think the Siegel Beach formula that Andreas will discuss today is really a beautiful powerful tool that you should all sort of be aware of okay thank you