 So, before I start, let me tell you that on the program schedule we've uploaded a handout for the lecture yesterday morning, where I talked about the three-gap problem. And in this handout is the full proof of the three-gap problem using the space of lettuces, which also was one exercise in the tutorial. So, you can download it and have a look at it and let me have your comments. So, today we're going, not today, but now we will talk about quasi crystals. And if you just recall, what we are interested in is a point process which is generated by a set P that is then randomly rotated and dilated by the matrix Kv, the rotation, and this diagonal matrix here. So, just as a reminder, this is simply the matrix 1 over T, T to the, which way around did I do this? Is it 1 over T or T in the top corner? So, it's correct, okay. And here we will have T to the 1, T minus 1, T to the 1 over T. Other way around, right? Anybody has their notes with me? It's not in my notes, so I can't look back. Is it T or 1 over T? T, exactly. Doesn't matter, it's late. Okay, so this is this matrix and this was the rotation matrix that rotated the vector V to the first coordinate. And we've seen, and what we are interested in, the big question is, starting with a fixed point set P, do we understand whether this point process converges to a limit? So, that's the big question. And what we have seen in the previous lectures is that when P is, for example, the cubic letters or any other Euclidean letters, then this works. And today, in this lecture, we will talk about when P is a quasi-crystal of a certain type. And I'll tell you exactly of what type it is. One example of the class of quasi-crystals that we will study is the famous Penrose tiling. So, of course, these are very famous. Just Google for it. And here's one nice example of a, one of the classical Penrose tiling. These are up-periodic tiling of the plane. And what we do to create a point set is we put a point at each vertex of this tiling. Yeah? And that will give us a point set. Now, how can we construct, for instance, the point set of the Penrose tiling? This is done by the so-called cut and project method. And how does this work? I will make a picture creating a one-dimensional cut and project set. But we're interested here in higher dimensional cut and project set. It's just a little bit easier to draw. So, what we will start off with, so the basic idea is this. You start with some lattice that is in higher dimensions. We want to construct here a one-dimensional point set. And we will have our physical space down here that, in general, will be Rd. In this picture, it's R. And we will use an auxiliary space, which is Rn. In this case, it's Rd. So, here we will have Rm. And n is equal to d plus m. This we will call the internal space. And this we will call the physical space. We choose a window set, which is a subset of Rm. And then we take all points that project orthogonally into this window set. These will be these points here. And then project those points onto the physical space. Again, orthogonally. So, we create in this way a point pattern. And if this lattice that we use in the construction is sufficiently generic, in particular, it shouldn't be aligned with a coordinate axis, etc., then you will get an up periodic point set. At the moment, I don't really care even if it's not generic, I might create a periodic point set. I'm happy with that for now. So, let me write down what I've described to you there. So, n is equal to d plus m. If m is 0, it's just going to produce a periodic point set. I will call pi the orthogonal projection onto Rd. I will call this the physical space. And pi internal, the projection, orthogonal projection onto the internal space, Rm, will be a lattice in Rn of full rank. Full rank doesn't mean I think Andreas, you've defined lattice to have full rank anyway. So, that will be a Euclidean lattice. And next A will be the projection of our lattice onto the internal space. And I'll take the closure of this. So, this will be an Abelian subgroup of Rm under addition. And A0 will be the connected component A that contains 0, contains the origin. So, this then is a linear subspace of Rm of dimension m1, let's say. So, the idea is the following here that when you start with a lattice and you look at the image of this projection, it could be dense in here. In this case, A would be simply equal to Rm. But it might not be dense. And usually in the quasi-crystal literature, this density is assumed because then you get the most interesting examples. But we don't want to do this. And the main reason is this gives a very nice construction of the Penrose tiling in this framework that we wanted to use, where indeed the projection is not dense. And when the projection is not dense, then the connected component will be not all of A, but rather the situation will look like this. So, you have your linear subspace. And then you have a direct sum over discrete subspaces. So, this is how it looks like. Now, we take mu A to be the harm measure on A. And we normalize such that if we restrict the harm measure to the linear subspace, A naught is the standard Lebesgue measure. Just a second. What's wrong? Oh, I'm sorry. Yes, this is a typo. Absolutely. This is pi interior because we are in the interior anyway. Absolutely. Very good. So, now what I want you to do is to simply forget about that, right? I just want you to ignore that fact and simply think of A as RM for now. This is just for the connoisseurs, if you like, the people who want to follow every little step, every little detail. And we are quite happy about this construction as it sort of allows us to unify several things in our work. But it's not so crucial for what I want to explain. Now, I haven't yet said what the window set is because that really defines the cut and project set. So, let me say what that is. So, the window set is a subset now of A. So, now the window set, let me call it window set, is now a subset of A. And what we'll assume usually is that it's bounded. And we also assume that the harm measure of the boundary set, of the boundary of the window set is zero. In this case, we say that we have a regular window and the corresponding cut and project set is regular. So, what is now our cut and project set? So, that's now the point set that we're interested in. That will be, for example, the point set of, that will be the vertex set of the, of the Penrose tiling for an appropriate choice of dimensions, projections and windows, dimensions, choice of lattice and windows. So, this will be, as I explained here, this will be now the blue points down here. So, the projections onto the physical space of all the lattice points that fall into the window set when projected onto the internal space. So, this is the definition. So, this is the whole construction, the whole cut and project construction. Are there any questions about that? Now, we have another little condition here that will make our life a little bit easier. We always will assume that, well, I shouldn't put that in brackets, that W and L are chosen in such a way the points of our lattice that fall into the window set are in, in one to one correspondence, I'm writing here, with our cut and project set. So, in order to achieve this, you could just make your window set a little bit smaller if you, if you have sort of, if, if a finite number of points would project onto the same point in P. So, we just want to rule out this thing so that we have a nice one to one correspondence between the points in here and in there, right? You could imagine that there are two points on top of each other, not in this one, but if the window set would be bigger and you had some points sitting on top of each other that would project onto the same, we want to rule this out. Again, that's a small technical condition. So, the first question we can ask when trying to understand the distributional properties of such a set is how many points are there in a big ball? And that's a classic result in the quasi-crystal literature which states that the number of points in such a cut and project set providing this is a regular set is proportional to the volume of the ball and Andreas will prove that formula tomorrow. Is that correct Andreas? Okay. So, that's our theorem one proved by Hof in 98, Schlotman also in 98 and then I think there's another name I forgot here and as we will see this is not, this just goes back to violet distribution so that doesn't require big machinery but it just was proved at this time because people were interested in this kind of question then. So, what do we do as before? We look at the number of points in a ball or of radius t, compare it with the volume of the ball, take the limit as t tends to infinity. I called this limit theta, so that's the definition of theta if that limit exists and the answer is that it's the harm measure of the window set divided by the volume of this thing here where v is the linear subspace r d cross a 0. So, this means the number of points in a big ball of radius t grows like the volume of that ball times this proportionality constant. So, in other words this is the density of that point, the asymptotic density and Andreas will show you how this works tomorrow. We'll take it as a given fact because now we want to come back to this question, can we show that this converges the point process that we get from p converges to a limit and what I will show you today is that we have to explain to you what is that brightness theory helps us to answer this question. And what we will do is as for in the case of lattice we will derive an equidistribution theorem for large spheres on the corresponding moduli space that will establish this convergence. Right, so the first task is I have to explain to you what is that moduli space. Let me clean the board first and as you see what I've just erased there is a lattice lurking in the background, the lattice L. Now this lattice is now a lattice not in d dimensions as we discussed before but in the higher dimension auxiliary space. So, it's not a surprise to you that the limiting moduli space that characterizes our point process will live on this big n dimensional space of lattices. And so we will now, okay let me call this g tilde, consider the group S L n r gamma S L n r gamma S L n r gamma S L n r gamma n z gamma tilde. And as before g will be S L d r and gamma will be S L do I need it? No let me take that away. So, S L d r will still feature because we're looking at a point set in d dimensions which we are rotating by a d by d matrix and dilating by d by d matrix, diagram matrix. Is there a question? What do you know? Here? Yes, aha, okay. So that's a very good question. I'm sorry that I sort of glanced over that. This is a linear subspace of R n. So it's basically R d, it's d plus m one dimensional. Just a, yeah. And so now I'm intersecting this subspace with a lattice L and it's a fact of the construction that the intersection of L with this V will be a lattice in V, okay? And that will become clearer tomorrow, maybe not. Okay, it will not become clearer anyway. It will become clear maybe you concentrate of just a not being R m, yeah. So just think of this being R m, okay? Then the intersection here will just be L, nothing will happen, okay? And so all we have here then is the volume of the fundamental domain of R n modulo L. Does that make sense? Right? So R n modulo L is a torus and that's just the volume of that torus in this case. And that's, before we looked at the letters of core volume one, I haven't said here that L is the letters of core volume one so it could be something else. And also in particular what could happen is that if this is indeed non-trivial then you could get other core volumes, yeah? Is that okay? Right. Now we have an action of D by D matrices and as it will turn out, these D by D matrices will act on this big space of letters in n dimensions. And so the right, there is a right way here for us to embed SLDR in G tilde and I'll call this embedding. So for a given G in G tilde and SLN R, I define this embedding which will turn out to be the good thing to do. So I'm embedding G in G tilde by simply taking my matrix A in SLDR and mapping it to the upper D by D blocks and then I knock in SLN R and then conjugating it by D. Okay. And now Retina's theorem already gives us the structure of in which cases this embedding leads to interesting subspaces in the space of n dimensional addresses. So Retina, Retina's theorem or theorems I should say imply that there exists a unique closed connected sub-spaces in the space of n dimensional addresses. Subgroup. So for every such little G here in G tilde and I'll call this HG such that the following properties hold. First of all, if I intersect, so this could be a very small subgroup, HG. And I'm intersecting now the letters SLNZ with this potential of essentially small subgroup. And the statement is that this is a lattice in HG. Second, note this is a characterization of HG, the things that I'm writing down here. Second, the image of the embedding of SLDR is contained in G. And third, if I consider the water orbit of my embedding. So, you know, I have an action now of SLDR on my space of n dimensional addresses G tilde modulo gamma tilde. And I'm looking at the orbit that's given by that particular embedding. And the third point here explains the relevance of this group HG for this dynamical interpretation. That is simply now the closure of this orbit is something nice. Namely, it's just the orbit of HG. Now, because gamma tilde intersect with HG is a lattice in HG, if you take HG modulo that lattice, it's a nice homogenous space and you can show that these two are isomorphic. And we are going to denote by mu G the ha measure on HG that becomes a probability measure on this quotient. And we'll also denote by the same mu G the corresponding probability measure on this space, yeah, which is the same under this identification. So, let me write this somewhere here. So, mu G is, no, let me write it a bit bigger so that you have it nice in your notes. Let me write it here. So, why is this important? Well, you remember we are rotating and dilating with points in SL DR this big lattice L. And now what this family of subgroups HG describes is basically all possible subspaces in the space of n dimensional lattices in which the dynamics can go, right? So, if we are inside one of those subspaces, we will never leave it by our SL DR action. Okay. So, mu HG or simply mu G is the unique HG invariant probability measure on this orbit. So, it's the unique probability measure when multiplying from the right with elements, all elements from HG. And this is just the harm measure on HG projected down and normalized. So, let me just say comes from harm measure, comes from harm measure on HG. So, trivial example, what would be a trivial example? Well, G tilde would be a subgroup that satisfy all these conditions. HG equals G tilde would satisfy all these conditions. Check it while I continue. Okay. So, remember what I had in mind here is to tell you something about a potential limiting process that we can describe in the case of lattices in d dimensions that Andreas and I discussed earlier. The limit was the space of lattices. And somehow the idea now is to construct an analog space of cut and project sets or space of quasi crystals that we can equip with a probability measure to get a point process that would be the psi that we are looking for. Okay. So, I need to now construct such a family of point sets that we can equip with a probability measure. And that probability measure will be exactly this harm measure here. Okay. So, let's pick G and G tilde and some delta such that the probability measure of the probability measure the lattice that comes up in the construction of our cut and project set can be expressed as Z n times G. And then here I'll put this scaling factor. So, delta is just some positive scalar. Before we didn't have this and the reason that we didn't have this before in d dimensions was that we always assumed L has co volume one. So, then when we act with S L the co volume is preserved. But now in the general construction of cut and project sets we haven't assumed this because in general we want to start with an arbitrary lattice and so this is just the scaling factor that will make this co volume one lattice to have the appropriate co volume here. And then what we show and I won't point out that I'll prove this to you here because we only have limited time. It's just something for you to believe and I'm happy at this late hour you believe everything I say. Right. So, what is important now we want to construct a family of such cut and project set. So, this is one cut and project set. Now I want to move this thing around. I want to move it around to create a whole family and I want to put a probability measure and then say that's my point process. Now what one can show, I write one can show OCS that if you move around this you put an H in here and so I can write my letters in this way for some G in G tilde. Now this G will define an HG over here. It's guaranteed by retina, guaranteed. We have it guaranteed. This HG exists by retinas theorem. This HG, this subgroup here, I now take this sort of set of lattices, this whole family of set of set is parametrized by HG and one can show that the physical space in which we live is preserved. Now I have told you don't worry about it. So just think of this as RM. In this case the statement is sort of trivial. Of course it's always an RM and the other statement that we prove here is that in fact it's equal to A for almost every H in HG and that's for almost every with respect to the harm measure on HG or the probability measure on here. So in this case would be the harm measure for all H. It's always the closure, yes. Well here we don't need it but here it is, correct Andreas. So what does this mean? Well it means that we now have a family of lattices and in the construction of our cut and project sets we don't have to change the A. That's a really good thing because we don't want to change the A because we want to keep the same window set. We don't want to have a space of quasi crystals. When we move around we also have to continuously change our window sets. That would not be very pleasant and so this little lemma here helps us to have this structure and so now we can define a map. All I say here by the way is in one of the papers on the reading list, we just highlight this reading list and this is a CMP paper, well it's the only CMP paper on the reading list. Communication is a mathematical physics and as far as we are aware this construction of families of quasi crystals is really new and what we can now do just as in the case of lattices we can define a map from this space here to cut and project sets, to the space of cut and project sets where we now take a cut and project set with the same window. So the fact that we can here use the same window almost everywhere is because A is the same almost everywhere and now we choose instead of the original lattice we choose the one that is translated by HG. So this map where we take a point in here, this is, oh that should be till this here, this you can think of parametrizing now these set of cut and project sets. So that's now the analog of what we had before where we took a point in G mod gamma and we established a bijection with the space of lattices. This now establishes a bijection of this space here at least for almost every H with these kind of cut and project sets. It's actually for every H because we could choose just the wrong window set, I mean that's okay I guess. We can just do it. We can take the window set in the larger space and just do that. The only thing is these will not be nice cut and project sets in general but that is good for almost every H. So what have I done? Well now this is my xi I claim. If I endow this space I make it a probability measure but simply now taking H to be random with respect to this probability measure on that space. So this is in, that's my claim and how do we prove this claim? Well we need to prove equidistribution as before. Now the equidistribution theorem will be more complicated and we will now indeed require Ratner's theorem for that. So theorem two. So let G be in G tilde, that's the same G as over there and then for any bounded continuous function f from ST minus 1 times HG to R we have that the following holds S. So we are now averaging this function over first coordinate is all the velocities that we are integrating over. The second coordinate is our embedded S and we have the following S. So we are now averaging this function over the first coordinate is all the velocities that we are integrating over. The second coordinate is our embedded SLDR rotation and dilation as before times. So as before lambda is a probability measure on the D minus one dimensional unit sphere that's absolutely continuous with the volume element on that unit sphere and what the theorem says is that as T tends to infinity this converges to the integral of f and okay. Why is that the good statement? Well it's the good statement for exactly the same reasons that we discussed before because now to prove that the point process converges to that psi T converges to psi what we need to show to prove convergent and finite dimensional distribution of these point processes is to pick here exactly the right characteristic function that says I want R points in my specific test set okay. And you can convince yourself that I've chosen the embedding exactly in the right way to act to respect this mapping here. So remember phi G was defined as the conjugation by G so you embed let me make a picture we have our n dimensional space of lattices so start with the cubic lattice and then what we do we rotate our cubic lattice by G to make it the lattice L that we are actually interested in and then we are acting with SLDR in a block form right because SLDR only acts on the physical space and leaves the internal space as it is so we have G times this block form I should do it like this G times this block form where here's the SLDR and then just to make it an embedding we have to also do G to the minus one to make it a group homomorphism okay. All right so by all that you've learned you hopefully believe me now that we can now show that theorem 4 makes sense. Is this theorem 4? No it should be theorem 3 right? Yeah 3 sorry. Is that now xi t converges to xi and let me just remind you xi t is again p kv dt with v random according to lambda absolutely correct. It continues with respect to omega. The only thing that I haven't told you the other ingredient is what? What else do we need to go from theorem 2 to theorem 3? We choose the right characteristic function fine what is the other ingredient that we need to make sure that our characteristic function has boundary of measure 0 or has small boundary if we approximate sets in the right way. I need some feedback I've been trained to ask for feedback and I forgot to throw out the entire course so now I need you to respond to me. What other important ingredient do we need to make sure that small sets small test sets lead to indicator functions here that have boundary of small that have small measure? Ah here it's heard something what? Siegelwitz yes. So Andreas today proved to you the Siegelwitz theorem under certain assumptions ok. So he said that the number of points in p should not grow too fast and we should have some uniform bounds on this etcetera etcetera. So proof theorem 2 plus Siegelwitz and Siegelwitz holds in this case so we can check all those in particular the we know how many points are in a big ball we in fact know how many points are in a big ball for each of those guys and so that's something Andreas will prove to you tomorrow that this counting function in fact works. Another really cool thing that Andreas will maybe explain to you tomorrow is that for some quasi crystals like the penrose tiling when you just look at the primitive point so you sit on the vertex and you look in certain directions and the penrose tiling you have the property that there are lines straight lines that contain infinitely many vertices and so you can ask the analogous question of the question of how many primitive lattice points there are it's the same here how many primitive vertices there are and there was no answer to that and Andreas and I provided the answer again using Siegelwitz backwards ok so that's very similar to what people do when they count closed geodesics on or saddle connections on flat surfaces where the counting of these is also a big question and there the Siegelwitz formula or approach also gives you that counting constant so that's something Andreas will explain to you tomorrow both counting all vertices in a big ball and counting just the primitive vertices in a big ball now what I haven't told you at all is so now we have an abstract formula for this limiting process right we have an abstract group HG whose existence is guaranteed by retina and we have a we have a harm measure on this group which we don't know what it is so what I want to tell you now is what examples can you get for HG in the last 10 minutes ok so let's see let's go over here yes so now this is this is a set for each H I get a set G is fixed right for each H I get a set H varies over this this space this space carries a probability measure so this is now a probability space and the push forward of this measure here under this map induces a probability measure on this set so this is now a random set or a random point process if you like P is fixed P here very good this P that we have over there is P of V delta 1 over N Z to the N G so that corresponds to the point H equal to 1 yeah that's the thing we start off with so that's our Penrose tiling that's the one we start off with then I've constructed this whole space with a nice topology and a nice probability measure on it that's the key point of what we are doing here and then this is of course a very important statement and that follows from retina's theorem in particular and I should say this it follows from a beautiful theorem corollary of Nime Shah to retina's theorem that deals with these kind of equidistribution problems in great generality and I know all this Andreas and I owe Shah's theorem a great deal okay right where was I wiping the board over here so now let me give you a few examples of G's that lead to particular H G's and if I manage I hope to even go to the Penrose tiling so the first proposition and as I said everything I'm telling you here you can find in the paper of Andreas and me and it is really a written for someone who is new to the subject we were in fact new to quasi crystals when we when we started with this M less than D and L is equal to Z D plus M G such that okay pi this is just some technical assumption what do I want to say here pi L is injective pi restricted to L is injective which is often a standard assumption then H G is in fact all of G tilde okay so if there is a problem with the dimension of your internal sprays is strictly less than D you never get anything else but the full G tilde so the limit will always be the full space of lattices and a nice example is something that was studied in the in the in the literature for the Lawrence gas was by Wenberg the so-called Fibonacci quasi crystals crystal where you take Q times Z which will be a point set an R2 periodic in one direction and this Q here is the following point set one plus tau squared plus one over tau square root one plus tau squared this is distance to nearest integer so you look at this sequence of points where J J runs over all integers and tau here is the golden ratio and this ratio so we call this the Fibonacci quasi crystal so you have a one dimensional set here that is exactly given by this cut and project for construction that I wrote down in the in the first in the first few minutes and here you see well this set you can construct from a letters in R2 projecting on a one dimensional space so what are the dimensions here we have a cut and project construction in R3 because it's you know art art times R2 and we project onto R2 so M is what M is one D is two so M is less than D strictly less than D so this applies so if you start with this set P here and you do our stuff you'll always end up in the space of lattices of dimension three now okay so we start with a two dimensional point set and our limit distribution is a cut and project set from a three dimensional random lattice now another very interesting construction and that's the one that's coming from that that will lead to the Penrose styling is the following and these are the most prominent as far as I can tell ways of constructing interesting quasi crystals they are based on number theoretic construction so K is a totally if you don't understand these words number field of degree so this is algebraic classical algebraic number theory and greater and equal to two over the rationals so an example would be you take the rationals and you adjoin square root two for example that would give you a degree two number field okay is the ring of integers in that number field so in this Q adjoin square root two it would be things that look like m plus square root two plus n m and n are integers and you can now define embeddings of this number field into the real numbers so these are n distinct embeddings and that allows you to represent an algebraic number as an R as a vector in R n and what you can then do is you can then associate with that a lattice by simply taking the embeddings of your integers in that number field and they will look like this so I'm embedding them in this way X is in O K and don't say anything Andreas I'll do this for pathological reasons hmm yes if it's a totally real number field that's in the assumption so you can always find that so now this would what be what this would be a n dimensional vector here but I actually don't want to just create an n dimensional lattice a dimension in n dimensions I want to create a lattice in d n dimensions so I am not just taking one number here I'm going to take a d dimensional vector and embed that component wise okay anyway it's a detail for those of you who are a little lost now at the end of the day just take for granted we'll start with a number field and that in a canonical way we construct the Euclidean lattice in this will be a lattice of full rank in R to the n d okay and now this R n d that will be my R n that I use for my construction so n is equal to n times d and to cut a long story short if we start with such a lattice here what will be H G B because in the end of the day all I need to tell you or you need to tell me the lattice that you want to take and here's a lattice that you get from this number theoretic construction and then you say well but what is the H G that will give me my nice probability space of cut and project sets and in this case one can work out that H G is G times S L D R S L D R S L D R times G to the minus one where there are exactly n copies of that. Okay and how do you get the G well the G right as before it's the same procedure we write the lattice L as some scaling factor times Z n times G and we read off the G from that. So that should be it for today what I have here on these two pages is and you can see it's very little the construction of the Penrose tiling using exactly this but I'll spare you that okay all I say is what you do here is you take as K this number field you choose a particular embedding and then you go through this construction and you choose a very particular window set that will lead to the vertex set of the Penrose tiling exactly following this construction. Now the groups that you get so remember what we had to do then is we had to take any S L n Z and intersect it with H G and what you will get if you do that you will get a Hilbert modular group and so the Hilbert modular group appear very naturally here as the groups that stabilize this family of cut and project sets or quasi crystals. Okay so what we're going to do tomorrow is two things the first thing is as promised Andreas will tell you how to prove that the number of points in a quasi crystal grows like the volume of the ball and then also the corresponding questions for the primitive points in a quasi crystal and then the last lecture tomorrow will be a more entertaining lecture it won't be on the board it will be about another application of our theory to graphs and networks namely circular graphs and how we can calculate the diameters in these graphs and make statements about their value distribution in the limit of large graphs as large random graphs in these particular families. So that will be hopefully a nice entertaining end of the lecture course so I'm really happy that still so many of you are here and so thank you very much.