Classification and statistics of cut-and-project sets
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Weiss, Barak
We introduce a class of so-called "Ratner-Marklof-Strombergsson measures". These are probability measures supported on cut-and-project sets in Euclidean space of dimension d>1 which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R) (affine or linear maps preserving orientation and volume). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patchcounting for typical cut-and-project sets.
 Thank you very much. I appreciate the opportunity to speak here. Please stop me at any moment if you're not able to understand what I'm saying or if you have questions or if you hear the police or any other thing like that. So this is a joint work with René Roux who is a postdoc at the Weizmann Institute and Yutamsimilansky who is a postdoc at Rutgers. And so I will begin the talk by sort of explaining roughly what we do without getting into details. Then I'll give a bit of a historical motivation and then I'll go back and get into more details. So the object of this discussion is so-called cutting project sets. Cutting project sets are a certain model for a periodic order. They're sometimes called or quasi crystals and they're defined as follows. You fix the following data. You fix dimensions dm and n where n is d plus m. You fix a direct sum decomposition of Euclidean n space where one sum end is called physical space and this is noted here and the other is called internal space there of dimensions d and m respectively. You have the corresponding projections related to this direct sum decomposition. L is either a lattice. All of you know what a lattice is or a grid. A grid is just a translated lattice and then you fix a window set W in the internal space and then you project this intersection to physical space. So the set you're interested in lives inside d dimensional space inside physical space. So here's a picture explaining this all the data that I had before still on this slide but now there's a picture. The picture is in two dimensions so n here is two d and m are both one. Physical space is this diagonal line running through the picture of one dimensional space. Internal space is the orthogonal line here. The current project set are the blue dots along the line so you're seeing that they're not a periodic set although they seem to have some weak kind of periodicity and that constructives follows. The black dots are the z2 lattice, the standard lattice. W is this red line on internal space so this pre-image of W here is this strip. You intersect the strip with the lattice that's the cutting part and the points that you get after cutting are visible here along this staircase type line and then you project all those points down to physical space and that's your construction. So this is a famous construction and well well studied and I'll say more about the history in a moment. The most famous example comes from the Penrose Tiling which has two alternative presentations or two acclimated definitions. I'll remind you not least because Penrose was recently a Nobel Prize recipient so we should remember one of his mathematical contributions. So here is the famous kite and the famous dart and they're decorated with these red and blue circular lines and you have a gluing rule which says that you can glue darts to kites edge to edge or to either to the each to the other or to themselves but you must make the red and the blue lines glue to themselves and so in other words I'm not allowed to reflect this dart and glue it here because then the red line wouldn't match up properly. If I were able to do that then I'd fill up a kind of a rhombus and this rhombus would tile space periodically and Penrose's discovery was that using these two pieces you can cover, you can tile the plane but only a periodically. So this is kind of a very nice example of an a periodic tiling and the connection to cutting project sets is that when you have this tiling by the kite and the dart or various related constructions due to this tiling then the vertex set, the points that you see as vertices of these tiles can also be presented as a cutting project set so that's a result of De Bruyne from the 1980s and here's a picture of the resulting tiling of space hopefully I can zoom in. So this is what you see locally in space and for the purposes of the lecture I just want to let me zoom in even further. You're going to see eventually the kites and darts appearing and you're also going to see the red and the blue lines matching up and what's important is that you have several different, quite different local pictures. So for example the picture that I'm putting a circle around will be called a patch later and it's rather different from the picture that I see if I circle around this point or around this point so there's a variety of local sceneries that you see and these are called the patches and they'll play a role later on in the lecture so we're interested in the geometry, large scale geometry of this type of discrete set and we're only interested in the vertex set in other words the discrete set itself and the tiling here is just to set things up and to make nicer pictures. So here are the questions that we want to answer in this project. We like to describe probability measures on collections of cutting project sets which are affine group invariant and ergodic and in this direction our main result is a complete classification of these measures so what do I mean by affine group invariant? So this is a probability measure so it's a way of randomly choosing one cutting project set out of a whole collection and it should have the probability that if I, the probability of choosing one particular kind of project set is the same as a probability of choosing its image under an affine transformation. What I mean by affine, the affine group is affine transformations of physical space which preserve volumes so a semi-direct product composed of translations and linear transformations with determinant 1 plus minus 1. And the second question that we want to understand is once you have fixed such a measure what is a typical cutting project set look like with respect to this measure? And in this direction what I say the words look like what I mean is I'd like to have results explain the frequencies of points inside certain large subsets of such a cutting project set and for both points and also the frequencies of various patches. So this is a kind of a typical question about the large scale structure of such a discrete set. So let me give a few more details about the spaces of discrete sets. We're going to be discussing measures on spaces of discrete sets so we're going to do that by first setting up the signal, the signal is going to be the Burrell signal with respect to a certain topology. And so what is this space and what is the topology? Curly C of Rd stands for closed subsets of Rd. It's a huge space made of lots and lots of sets. All closed subsets are points in this space and we can topologize it with a certain metric and then which I'll describe in a second. And then once we have a discrete set we can look at its closed orbit closure under translations. So every set can be translated and we can make orbit closures with respect to a natural topology and these kinds of set omega are called tiling spaces in literature and they're dynamical systems with an Rd action. And to explain the topology here is the definition of a certain metric called the sometimes called the shovel T field metric. The distance between two points are close to each other. So the points the sets here are y0 and y1. They're both closed sets. 1 minus i here just flips between 0 and 1. So y0 is close to y1 if when you intersect it with this large ball around the origin epsilon is a small number. So this is a large ball. You take the intersection of y1 with a large ball around the origin. Every point you see is within epsilon of a point of y0. And this also holds when you flip the rolls of y1 and y0. Okay, so this is a kind of an obvious natural metric on spaces of sets. And if you look at these space of closed sets with this metric it becomes a compact metric space. And the closure operation that I'm mentioning here could be taken with respect to this particular metric. The metric itself doesn't play an important role in what I'm going to say but it's the easiest metric to define. And when sets are nice, for example, discrete sets with certain properties like finite local complexity and so on, many of you are probably familiar with those settings. This topology coincides with the other topologies that people use in the literature. So this is not an exotic topology. It's the natural topology to use. As I said, I welcome questions if there are any. So please stop me whenever you have a question. So in the literature, so I'm going to be discussing renormalization by the action of ASLDR. As I said before, ASLDR is the group of affine measure preserving transformations. But typically in this business, the renormalization that's usually used is by dilations, by the positive real numbers acting by dilations. And the importance of this dilation action, I just want to review that slightly quickly, is that it renormalizes the dynamics by translation on the tiling space. So here's the idea in a nutshell. So here's a way, a kind of classical way to construct tiling. This is the notion of the substitution tiling. We have a collection of three triangles with three colors. And we have a rule which replaces each one of the triangles with a bigger triangle inflated by some fixed inflation constant and subdivided back into the original shape. So here we see the big tile, the big tile corresponding to the green, to the dark green is partitioned into two copies of the dark green triangle, two copies of the yellow triangle, one copy of the light green triangle. And when you have this kind of rule, you can use it to construct a tiling. Here's a tiling that you can obtain by applying this rule repeatedly. So it's a part of a tiling of the infinite plane. And this tiling is naturally hierarchical. So you can think about dilating it, which has the effect of zooming out. So you look at it from further and further away. And because of the tiling rule, the edges are going to, if you remove some of the edges and keep others, you're going to see the same tiling with the same tiles on a bigger scale. So in a sense, this dilation dynamic, or this zooming in dynamics, will have recurrence. And when you have recurrence for the dynamics of zooming in, that gives you a lot of lots and lots of information about the dynamics of the translation action. That's one way to prove unicardicity and many other counting formulas and many other related things. So this is sort of the classical point of view of zooming in dynamics or dilation dynamics for renormalizing in order to understand the tiling spaces that are related to substitution tiling. But for cutting product sets, we do something else. And this is the ASLDR renormalization. So we're interested, so as I said, ASLDR is the semi-direct product of SLDR, the determinant one linear transformations of r to the d, and the translations of r to the d. And you can use this group to renormalize the space of closed subsets. Obviously, this group acts on rd, so it acts on closed subsets of rd. And if you understand the dynamics of this group, you're going to be able to understand individual point sets, and you're going to be able to understand the geometry and the rd action on these tiling spaces. So this is a relatively recent point of view in the literature. It was first used by Markloff and Strombergson in a sequence of very beautiful papers. So I left out the words very beautiful, but they are really beautiful papers of Markloff and Strombergson. And they use this dynamical system to address certain questions in statistical physics, in mathematical physics. I also want to mention some of my own work with Solan and Solomon. And we also use this kind of renormalization or this kind of dynamics in order to get partial results on the Dancer problem. So the Dancer problem is a wide open classical problem in discrete geometry from the 50s. We haven't solved it. It's still widely open, but it's a beautiful question. So let me mention it for those of you who are not aware of it. The answer is asking, does there exist any set in r to the d, which is a discrete set, the number of points of ball of radius t grows up to constants like the volume of the ball. And the set, so this means there's not too many points in the set. Somehow. But on the other hand, there's sufficiently many points so that this set intersects non-trivial every convex subset of rd of unit volume. This is a wide open question. It's been open for a long time. We haven't solved it, but we've shown that one way to attack it is to understand this action on the space of close upsets of r to the d. If you had such an example, maybe you don't believe that such an example exists. If such an example did exist, its orbit closure under this group would have very strong properties that sort of the, this action detects, specifically detects this property of intersecting every convex set of volume and since precisely because this group of affine transformations preserves the collection of convex sets of volume one. Okay, so this was just a kind of an advertisement for this ASLDR renormalization direction of study. And specifically, Mark Lofen-Strombergsen asked two questions about this ASLDR dynamics. First question is, can we describe the invariant measures which are supported on discrete subsets? So as I said, this collection contains all closed subsets. The discrete subsets are some kind of a subset of this space and they want to know which invariant measures there are. What can put on this collection? And a more refined question, given such a measure, mu, and given a point in its support, is the following true? So what is this statement? Maybe it's easier to parse this information when you think about the SL2 case. So k lives in the group SLDR, so you might take it to be the group of rotations. So this is just a circle. We're taking our set and we're rotating it through a circle's worth and that's giving us a measure. And then we push the measure by diagonal matrices. And so the circular action becomes a kind of a long ellipse. And we're sampling the collection of closed subsets by performing this operation to our fixed base point, y. And we want to know whether it samples mu fairly. It's a kind of an ergodic theorem for a big group, the group SLDR, with these kinds of averages. And we want to know whether there is such an ergodic theorem and for which specific points does it hold. So this is definitely a difficult and technical problem in dynamics and it turns out to be a crucial problem for the applications in statistical physics that Mark Loughlin-Strombergsson are interested in. And so you might wonder if there are any such measures satisfying these properties. And Mark Loughlin-Strombergsson showed that they are, that they are and they are supported on, the discrete sets that they're supported on are cutting project sets. And this, their construction relies on homogenous dynamics and specifically it relies on a very famous body of work by Ratner from the 1990s describing the, doing the measure classification for actions of groups generated by unipotence. Ratner's work is used twice. The first place it's used is just to construct the measures. You're going to see that in a moment. And the second place it's used, which I won't talk about today, is to prove this, understand exactly this kind of equidistribution phenomenon. So here is the construction of Mark Loughlin-Strombergsson. And since it's introduced by Mark Loughlin-Strombergsson, it relies strongly on the work of Ratner. We're calling it, in this joint work, we're calling it the Ratner Mark Loughlin-Strombergsson measures and it's going to be, the abbreviation is going to be RMS. So these are what we call RMS measures. So what do you do? You fix the dimensions of your cutting project construct. This is a measure supported on cutting project sets. You fix the dimensions in the cutting project constructions. You fix the direct somebody composition, physical space and internal space. You fix the window. So the only data in the construction that I have not fixed here is the lattice that was used. The lattice L or either the lattice or the grid. Now, with this data fixed, you consider the following embedding of ASL-VR, affine maps of D-dimensional space into ASL-NR, affine maps of N-dimensional space. Here's the algebra of the embedding. V is the translation, G is the linear transformation. What you should think about geometrically is inside R to the D, sorry, inside R to the N, you have physical space as a D-dimensional subspace. And you just let the affine map move things in physical space without touching internal space at all, acting true for the N internal space. So that's the meaning of this embedding. Once I've fixed this embedding, I can consider the space of grids. This is a homogeneous space for this group ASL-NR. And on the space of grids, suppose I could find any measure, which I'm going to call U-bar, any probability measure, which was invariant under the ASL-D action, this group via this embedding is acting on N-dimensional space because these are N by N affine transformations. And so I take such a measure, U-bar, on the space of grids, such measures are associated with certain groups H, which are intermediate between ASL-D and ASL-N. That's the important work. That's the RACNOR measure classification theorem. And given such a measure, U-bar, what you can do is you can project it to a measure on cutting project sets because you have a map that says your grid, L is the grid, L is an element of the space, to the set lambda LW, which is the cutting project set constructed from this data, including W, and from the grid as in my first slide. Okay, so this is an important construction. I hope I've made it clear what the construction is. If there are any questions, as I said, please stop me. So this is the construction of Markliffe and Strombark Center. And I want to sort of give you a teaser of the things we proved first and then make mention of conjecture. So the first thing we prove is that any measure, which is ASL-D, invariant and ergodic, and if suppose the measure gives full less to cutting project sets is one of the measures I just described. There's no other way to put a measure which is affine and invariant and ergodic and with probability one produces a cutting project set. And there's a technical condition on the cutting project sets. They should be irreducible. Irreducibility for the experts in the audience means it's a standard assumption in the topic of cutting project construction. So irreducibility means that for almost every L, the projection onto internal space is dense and the projection onto physical space is injected when restricted to L. And so that's our first theorem. And I want to just mention a question. A question is, is there any other interest in construction of a measure, not just on cutting project sets, but more generally on Dillene sets. A Dillene set is a set that's uniformly discrete. So the distance between any pair of points is at least some positive number and it's uniformly dense. There are no arbitrarily large open patches without any points of your set. So my conjecture is that any ASL-D, or invariant and ergodic measure, which assigns full measure to Dillene sets, again, Dillene sets is a much wider class than irreducible cutting project sets. Any such measure, well, I wrote has an untrivial joining with an RMS measure. Just as a first approximation of the conjecture, just say, just ignore the words joining. Is it true that any, can you, is it true that any such measure is actually the measure that is described on the previous slide? Now, that's not exactly true. As I just stated it, what you can do is you can take a measure that's constructed as in my previous slide and then randomly, in a smart way, you can randomly add more points without destroying this uniform discreteness property and without destroying the invariance and ergodicity under ASL-DR. That can be done, but that's somehow cheating. In a sense, I started with a measure which did come up according to my construction, to the construction I just outlined. Again, so this discussion about the joining is a statement for which I have no counter example. That's the conjecture. The only way to construct such measures is using the construction I just outlined. Now, I don't have very strong evidence for this conjecture. The only evidence that I have is that I tried several times using several different ideas to construct such a measure, which is supported on this collection. You should remember that this ASL-DR group is a big group. I wasn't able to. If you think, I'd be happy to be proved wrong if you have a counter example, but I find it challenging to envision any counter example to this conjecture. That was basically giving you an idea of what we care about and why. I think this conjecture, if proved, would have very strong consequences for both problems in mathematical physics, for the problem, the answer question that I discussed, and other problems about discrete geometry. I don't think it's a curiosity. I think if it's true, it's quite a strong result with a lot of consequences. Having said that, let me move on to our results, which are not yet at all close to proving this conjecture, but this was motivating the things I'm interested in. Now, I want to talk about our results in more detail. For now on theorems, we'll be numbered. Let's just point that out. If a theorem is numbered as it is over here, that means it's one of the theorems in the present paper. The second theorem is a classification of the RMS measures, meaning Mark Lofen-Strombergsson said that the measures could be described using Ratner's theorem, but they didn't actually mail all of the data that comes up in Ratner's theorem. The data in Ratner's theorem involves closed orbits for certain groups, and we give a list of the groups. That's the content of the theorem too. Whenever we have an S of the R-Immvariant measure, we know by Ratner's that it's H-Immvariant and support on a closed H orbit. Here is H. It's given by our theorem. H has a structure of a semi-direct product. In the semi-direct product, the group of translations is the entire group of translations in Rn. It can't be any smaller. The semi-simple part, H prime, has a description in terms of restriction of scalars from a group defined over Q. These are notions from the theory of algebraic groups, and I'm not planning to go into them in detail, but what you should notice is that they're only two cases, and they give you strong conditions on the relationships between D, which is the dimension of physical space, and N, which is the dimension of the total space. One of the two cases, the symplectic case, is only possible when D is two, when the dimension of space is two. Most, somehow, if the dimension is larger than two, the only case that can arise is coming from SLK. I'll give you an example now. Here are the three examples that can come up in the simplest non-trivial case, which is when D is two, and is also two, and therefore N is four. The first case is, somehow, the big measure, the measure of biggest support. The group H is the entire affine group of four dimensions, SL4, semi-direct R4. The measure is the canonical ASL4R invariant measure on the space of four-dimensional grid, which I'll denote by this M of YN. N is four here. That's the first case. That's the big measure, so-called. There's an intermediate group, an intermediate measure, which lives on block matrices. These are four-by-four matrices, and they're being divided into two-by-two blocks. These are associated with real quadratic fields. I'm giving you an explicit example, an explicit construction of the grid or the lattice that will appear. If you want to use this group, you can't just use it. You need to find a suitable lattice, and this is the lattice. It's very, very number theoretical. It involves this conjugation in this quadratic field, and the Penrose vertex set can be written in exactly this form. In the Penrose case, it's exactly case two. The third example possible in this dimension is the symplectic case. The group H is the symplectic group, and the lattice is a symplectic lattice. It has a basis, which is a symplectic basis of four-dimensional space. This is the sort of an example of how the classification works in dimension two with N equals to four. Now I'm going to give you a crash course on the Ziegels summation formula, which is what is the Ziegels summation formula? The Ziegels summation formula is a certain formula that was developed for studying lattices. What we're able to do in our project is to prove analogous results for the measures on lattices for these RMS measures, which we've just introduced. We have a bunch of formulas, which I want to describe. They're all analogs of work done, all classical work done in the 40s and 50s. Here's the classical Ziegels summation formula. Currently X of N here is the space of lattices of covalent one. Given a function on Rn, you form a new function on the space of lattices whose value on a lattice L is the summation over all lattice points, all non-zero lattice points of the value of the original function f. For example, if f were the indicator of a ball, this would be just counting the number of points in the intersection of the lattice with the ball. The Ziegels summation formula from 1945 says that after normalizing by a constant c, we have this beautiful identity. The integral of the original function f on Rn is the same as the integral of the transform of f hat on the space of lattices. This is a quite important formula. It's used because the measure on the space of lattices is quite hard to work with in practice. The easiest way to work with it is through this identity. One of the beautiful things done by Markloff and Stromberg's in 2014 is that they extended this formula replacing the measure on the right, the measure on the space of lattices with an arbitrary RMS measure mu. Where the transform instead of summing over all non-zero lattice points, you sum over all points in your cutting project set. As you see, I've written down the Markloff-Stromberg's result in this purple color. For now on the results which are analogs of classical results, classical results will be written in black and their analogs will be written in this purple color. This is the Markloff-Stromberg's and analog of the Ziegels summation formula. A further classical summation formula is a higher dimensional one. I've copied the original Ziegels summation formula here and I'm defining a new higher dimensional analog. You fix another parameter p and then instead of summing your function on vectors in your lattice, you sum over p-touples of p-different vectors in your lattice and therefore the function f depends on p times n numbers which I think of as p vectors in R to the m. This is the p-th Ziegels transform of your function f evaluated on the lattice. There's an analogous formula where you recover the integral of this p-th Ziegels summation from certain obvious measures on Rn to the p. For example, the Lebesgue measure on Rn to the p or for example, measures supported on pairs of linearly dependent vectors or p-touples with a certain kind of integer linear dependence and so on. There's a kind of an explicit formula which is too complicated to write down which was proved by several people and that's the analog of the Ziegels summation formula. But before stating the Ziegels summation formula, one more thing you want to know is is this function in L1, is it integrable? Even integrable just to get the conversation started. The first results in this direction are due to Rogers and Schmidt from the 60s and strengthening is due to Eskin or Gullisonosis. I didn't have room on the slide for their names so that's emm. Here's what they prove. If p is an integer less than n and is the dimension then this transform is in L1. Sorry, yeah, then this transform is in L1. A sharper result is if you take any q smaller than n, so q need no longer be an integer, it could be any real number smaller than n and you look at the original Ziegeltransforms written here, then its integrability exponent, it's in Lq. This is sharp. So the largest, so for any q less strictly less than n, f hat belongs to Lq and as soon as q becomes n, I didn't write that on the slide, as soon as q becomes n, f hat is no longer going to be in Ln. That's more or less classical except for this new contribution from the 90s and we get the analog. So we have the same two results for integrability balance of the transform in the case of cutting project sets except the parameter n that appears here and here has to be replaced with this number 1 plus the q-rank of the group H prime that I had in the previous slide. So that's a algebraic group theory invariant associated with this and as an interesting corollary what you can see now is that if I fix the dimension d and I use different measures involving bigger and bigger n, I can make this quantity larger and larger. In particular it means that for larger and larger d, I can find analogs of this Zika summation formula with better and better integrability bounds for the transform. That's an interesting fact that's not, can't be done for lattice. We have more flexibility for cutting project sets. As I said earlier, once you have this p-th transform, you can try to write down an identity for it and so this was done in the 50s and 60s by Rogers, Schmidt and Andreve in separate work and as I said they have combinatorial formulas for expressing this quantity and we have the analog for rms measures. We have kind of an infinite sum which expresses these expressions and they also give us an expression for the higher moments of the original Zika transform and the shape of the measures that appear is the flavor is given by kind of a cutting project type construction for the big measure rather than for the discrete measures. Another thing we prove is something called the Rogers second moment bound. So for lattices in the 50s, Rogers proved the following fundamental fact. The this expression which is the second moment of f-hapt, if you fix f and you think of this as a random variable depending on the choice of the lattice lambda, then this is the second moment and Rogers bounded the second moment in terms of the space into root of f which is as we saw earlier in the Zika transform. This is actually the first moment, it's the expectation. So the second moment is bounded by a constant times the expectation. That's a very important input to counting results and we prove the analog theorem in the full context of an arbitrary rms measure. And using this as an input, we're able to do something that Schmitt did in the 60s for lattices. So what did Schmitt do for lattices? You start with any collection of sets which is which we call an unbounded ordered family. So what this means is the sets are parameterized by positive real numbers and when you have two, if you have t1 less than t2, then the corresponding sets are contained in each other. So an example to think about is just a grand collection of balls, Euclidean balls, but this could be a much more complicated example. You could you know increase, take kind of a stripped very larger and larger or some kind of spiraling bodies or whatever you want. It's just an unbounded ordered family of sets and all you require is that they're each individually refining volume but their volume is increasing to infinity. And in this setup, Schmitt discussed the typical growth of the point counting problem in these sets. So again, just think of balls as a first approximation. So the left hand side is the number of points that your lattice has with the large ball, say a ball of radius t. And Schmitt said the main term grows at the volume which was known before and follows from the Ziegels summation formula. And Schmitt established an error term which is on the order of the square root of the volume. So the quality of this error term, the fact that we have a power savings, that's what I mean here by effective counting. It's counting with an explicit bound on the error rate. This result of Schmitt from 1960 has not been improved. Even if you talk about Euclidean balls, not just this general, very general setup of an unbounded ordered family, just take Euclidean balls and count the number of lattice points in Euclidean ball. If you want to know the behavior of the error term for the counting problem in a typical lattice, typically with respect to the natural measure on the space of lattices, that's an open problem. And Schmitt's bound is the best bound known to date. There's a conjecture of Goetze that this can be improved by reducing two by subtracting one over two d, if I remember correctly, in the exponent. But that's an open problem and Schmitt's result is the best possible. And we prove the pattern project analog of this result. So we prove an effective point counting for the RMSH measures. We have the same exact error bound because we use the same technique. And this involves these Rogers second moment bounds. With the second moment bounds, we're able to look at the exact same asymptotic. And I wrote down here what I said earlier, it matches the best known almost every bounce, even for the special case of lattices and Euclidean balls. Now, this is an almost everywhere result. It's telling you the counting behavior of sets and balls for almost every lattice or for almost every kind of project set. But it doesn't tell you what's going on for a specific one. If you care about a specific one, say the Penrose example or other examples, then there are other techniques. So for example, the Penrose example is better understood using Perron-Ferbenius theory and the tiling, the substitution tiling machinery. This was done in a paper of Solomon, Gar-Solomon. If you talk about the kind of project instructions, you can get a better handle using Fourier techniques and diaphanthine approximation for certain explicit lattices. I mean, the data and the kind of project construction should have certain diaphanthine properties. And then you can get certain effective counts. They're not better than this, but they are explicit. I mean, you have explicit examples of cutting project constructions. So this is joint work that I did with Haynes and Kelly. And there's been other work using Panamikri variant chronology on tiling spaces by Schmiedt and Trevino for cutting project sets that have a certain dilation, self-similarity under dilations. Okay. Another thing that we are able to do is to count patches. So I showed you earlier, I want to show the slide of the Penrose tiling, what I mean by a patch. So a patch is you take your original point set, you centered it. If it were instead of you move the center from point X to the origin, and then you look at the intersection with the ball of radius R, and this is called the R patch centered at X. And obviously, if R is fixed, you may see the same patch again and again at different Xs. And you might want to count the appearances of this patch again in a large ball. And this counting problem had only been considered up to the point of proving that there was a main term in the asymptotic. So there was a kind of a prescribed order of growth. And this is sort of linked to the question of unique ergodicity. And if you want to get an error term, in other words, if you want to get effective counting, it's kind of like a form of quantitative unique ergodicity for these tiling spaces. And we're able to do that for again for almost every almost every cutting project set where almost every is with respect to any RMS measure. So fix an RMS measure, fix any window which satisfies a certain property of small boundary in the sense of upper box dimension. Then you can find a power savings parameter, which is this theta, such that for any patch, you have this order of growth of the catch of the patch counting. There's a main term here. And then there's the error term. And the implicit constants in the error term are going to depend on on the data. In other words, they depend on dealt on the land on the kind of project set. They depend on the window. They depend on this dimension of the window of the boundary window. And we get a power savings. And I just wanted to point out that this result has no analog for that. It says simply because for analyze, it doesn't make sense to cut patches. Every point has the same patch around it. There is no analog question. So this is the first effective counting result for patches that we are aware of. Okay, I see that I managed to put everything in my time. And that's it for me. Thanks for your attention and stay healthy. Yeah, so thank you very much for the very interesting talk. So it's open for discussion. We have some minutes time. So to start the discussion, just a small question from me. The constant C lambda in the Schmitt like theorem. What is known about that? So they have an explicit formula. You have an explicit. It's not our work. This is the main term constant was done by several people. I don't remember the names right now. One of them being moody, but there was earlier work. And it's expressed in terms of the co volume of the lattice L and in terms of the volume of the window. It's just simple formula. Thanks. Yeah, Nikolai, our next speaker wants to ask a question. So please, you can ask the question I can open. Yeah, you can. Just one question. So I would like to ask if the results about classification of your measures lead to certain mathematical corollaries of such a type. If a constant project set is self similar, then something is algebraic. Well, they should. And but it's in this kind of thing has been already done by mayor. So may have this beautiful theory from the 60s, where he's proving that if you have a what we call the kind of project set, may have caused it a model set. If you have a model set, and it has the property that when you apply a dilation, the model set becomes the subset of itself. This he calls this a similarity. Then the coefficient is algebraic integer and mayor gives a lot of restrictions on exactly the algebraic integer that arises. It's part of his beautiful, beautiful theory. We are not interested in these dilations. We're interested in self maps, which come from the affine group. And in principle, the answer is yes. I mean, it should be detectable. These kinds of things should be detectable from the algebraic data. We haven't done it so far, but I think there's plenty of plenty of things of that type, just which can be done. Yeah, are you satisfied, Nikolai? Yes. Yes. Okay. Thanks for that. He can't help saying he's satisfied because he's the next speaker and I could ask him a question. Yeah, of course.
 Thank you very much. I appreciate the opportunity to speak here. Please stop me at any moment  if you're not able to understand what I'm saying or if you have questions or if you  hear the police or any other thing like that. So this is a joint work with René Roux who  is a postdoc at the Weizmann Institute and Yutamsimilansky who is a postdoc at Rutgers.  And so I will begin the talk by sort of explaining roughly what we do without getting into details.  Then I'll give a bit of a historical motivation and then I'll go back and get into more details.  So the object of this discussion is so-called cutting project sets. Cutting project sets  are a certain model for a periodic order. They're sometimes called or quasi crystals  and they're defined as follows. You fix the following data. You fix dimensions dm and  n where n is d plus m. You fix a direct sum decomposition of Euclidean n space where one  sum end is called physical space and this is noted here and the other is called internal  space there of dimensions d and m respectively. You have the corresponding projections related  to this direct sum decomposition. L is either a lattice. All of you know what a lattice is  or a grid. A grid is just a translated lattice and then you fix a window set W in the internal  space and then you project this intersection to physical space. So the set you're interested  in lives inside d dimensional space inside physical space. So here's a picture explaining  this all the data that I had before still on this slide but now there's a picture. The  picture is in two dimensions so n here is two d and m are both one. Physical space is this  diagonal line running through the picture of one dimensional space. Internal space is  the orthogonal line here. The current project set are the blue dots along the line so you're  seeing that they're not a periodic set although they seem to have some weak kind of periodicity  and that constructives follows. The black dots are the z2 lattice, the standard lattice. W  is this red line on internal space so this pre-image of W here is this strip. You intersect  the strip with the lattice that's the cutting part and the points that you get after cutting  are visible here along this staircase type line and then you project all those points  down to physical space and that's your construction. So this is a famous construction and well  well studied and I'll say more about the history in a moment. The most famous example comes  from the Penrose Tiling which has two alternative presentations or two acclimated definitions.  I'll remind you not least because Penrose was recently a Nobel Prize recipient so we  should remember one of his mathematical contributions. So here is the famous kite and the famous  dart and they're decorated with these red and blue circular lines and you have a gluing  rule which says that you can glue darts to kites edge to edge or to either to the each  to the other or to themselves but you must make the red and the blue lines glue to themselves  and so in other words I'm not allowed to reflect this dart and glue it here because then the  red line wouldn't match up properly. If I were able to do that then I'd fill up a kind  of a rhombus and this rhombus would tile space periodically and Penrose's discovery was that  using these two pieces you can cover, you can tile the plane but only a periodically.  So this is kind of a very nice example of an a periodic tiling and the connection to  cutting project sets is that when you have this tiling by the kite and the dart or various  related constructions due to this tiling then the vertex set, the points that you see as  vertices of these tiles can also be presented as a cutting project set so that's a result  of De Bruyne from the 1980s and here's a picture of the resulting tiling of space hopefully  I can zoom in. So this is what you see locally in space and for the purposes of the lecture  I just want to let me zoom in even further. You're going to see eventually the kites and  darts appearing and you're also going to see the red and the blue lines matching up and  what's important is that you have several different, quite different local pictures.  So for example the picture that I'm putting a circle around will be called a patch later  and it's rather different from the picture that I see if I circle around this point  or around this point so there's a variety of local sceneries that you see and these  are called the patches and they'll play a role later on in the lecture so we're interested  in the geometry, large scale geometry of this type of discrete set and we're only interested  in the vertex set in other words the discrete set itself and the tiling here is just to  set things up and to make nicer pictures. So here are the questions that we want to  answer in this project. We like to describe probability measures on collections of cutting  project sets which are affine group invariant and ergodic and in this direction our main  result is a complete classification of these measures so what do I mean by affine group  invariant? So this is a probability measure so it's a way of randomly choosing one cutting  project set out of a whole collection and it should have the probability that if I, the  probability of choosing one particular kind of project set is the same as a probability  of choosing its image under an affine transformation. What I mean by affine, the affine group is  affine transformations of physical space which preserve volumes so a semi-direct product  composed of translations and linear transformations with determinant 1 plus minus 1. And the second  question that we want to understand is once you have fixed such a measure what is a typical  cutting project set look like with respect to this measure? And in this direction what  I say the words look like what I mean is I'd like to have results explain the frequencies  of points inside certain large subsets of such a cutting project set and for both points  and also the frequencies of various patches. So this is a kind of a typical question about  the large scale structure of such a discrete set. So let me give a few more details about  the spaces of discrete sets. We're going to be discussing measures on spaces of discrete  sets so we're going to do that by first setting up the signal, the signal is going to be the  Burrell signal with respect to a certain topology. And so what is this space and what is the  topology? Curly C of Rd stands for closed subsets of Rd. It's a huge space made of lots  and lots of sets. All closed subsets are points in this space and we can topologize it with  a certain metric and then which I'll describe in a second. And then once we have a discrete  set we can look at its closed orbit closure under translations. So every set can be translated  and we can make orbit closures with respect to a natural topology and these kinds of set  omega are called tiling spaces in literature and they're dynamical systems with an Rd action.  And to explain the topology here is the definition of a certain metric called the sometimes called  the shovel T field metric. The distance between two points are close to each other. So the  points the sets here are y0 and y1. They're both closed sets. 1 minus i here just flips  between 0 and 1. So y0 is close to y1 if when you intersect it with this large ball around  the origin epsilon is a small number. So this is a large ball. You take the intersection  of y1 with a large ball around the origin. Every point you see is within epsilon of a  point of y0. And this also holds when you flip the rolls of y1 and y0. Okay, so this  is a kind of an obvious natural metric on spaces of sets. And if you look at these space  of closed sets with this metric it becomes a compact metric space. And the closure operation  that I'm mentioning here could be taken with respect to this particular metric. The metric  itself doesn't play an important role in what I'm going to say but it's the easiest metric  to define. And when sets are nice, for example, discrete sets with certain properties like  finite local complexity and so on, many of you are probably familiar with those settings.  This topology coincides with the other topologies that people use in the literature. So this  is not an exotic topology. It's the natural topology to use. As I said, I welcome questions  if there are any. So please stop me whenever you have a question. So in the literature,  so I'm going to be discussing renormalization by the action of ASLDR. As I said before,  ASLDR is the group of affine measure preserving transformations. But typically in this business,  the renormalization that's usually used is by dilations, by the positive real numbers  acting by dilations. And the importance of this dilation action, I just want to review  that slightly quickly, is that it renormalizes the dynamics by translation on the tiling  space. So here's the idea in a nutshell. So here's a way, a kind of classical way to construct  tiling. This is the notion of the substitution tiling. We have a collection of three triangles  with three colors. And we have a rule which replaces each one of the triangles with a bigger  triangle inflated by some fixed inflation constant and subdivided back into the original  shape. So here we see the big tile, the big tile corresponding to the green, to the dark  green is partitioned into two copies of the dark green triangle, two copies of the yellow  triangle, one copy of the light green triangle. And when you have this kind of rule, you can  use it to construct a tiling. Here's a tiling that you can obtain by applying this rule  repeatedly. So it's a part of a tiling of the infinite plane. And this tiling is naturally  hierarchical. So you can think about dilating it, which has the effect of zooming out. So  you look at it from further and further away. And because of the tiling rule, the edges  are going to, if you remove some of the edges and keep others, you're going to see the same  tiling with the same tiles on a bigger scale. So in a sense, this dilation dynamic, or this  zooming in dynamics, will have recurrence. And when you have recurrence for the dynamics  of zooming in, that gives you a lot of lots and lots of information about the dynamics  of the translation action. That's one way to prove unicardicity and many other counting  formulas and many other related things. So this is sort of the classical point of view  of zooming in dynamics or dilation dynamics for renormalizing in order to understand the  tiling spaces that are related to substitution tiling. But for cutting product sets, we do  something else. And this is the ASLDR renormalization. So we're interested, so as I said, ASLDR is  the semi-direct product of SLDR, the determinant one linear transformations of r to the d,  and the translations of r to the d. And you can use this group to renormalize the space  of closed subsets. Obviously, this group acts on rd, so it acts on closed subsets of rd.  And if you understand the dynamics of this group, you're going to be able to understand  individual point sets, and you're going to be able to understand the geometry and the  rd action on these tiling spaces. So this is a relatively recent point of view in the  literature. It was first used by Markloff and Strombergson in a sequence of very beautiful  papers. So I left out the words very beautiful, but they are really beautiful papers of Markloff  and Strombergson. And they use this dynamical system to address certain questions in statistical  physics, in mathematical physics. I also want to mention some of my own work with Solan  and Solomon. And we also use this kind of renormalization or this kind of dynamics in  order to get partial results on the Dancer problem. So the Dancer problem is a wide open  classical problem in discrete geometry from the 50s. We haven't solved it. It's still  widely open, but it's a beautiful question. So let me mention it for those of you who  are not aware of it. The answer is asking, does there exist any set in r to the d, which  is a discrete set, the number of points of ball of radius t grows up to constants like  the volume of the ball. And the set, so this means there's not too many points in the set.  Somehow. But on the other hand, there's sufficiently many points so that this set intersects non-trivial  every convex subset of rd of unit volume. This is a wide open question. It's been open  for a long time. We haven't solved it, but we've shown that one way to attack it is to  understand this action on the space of close upsets of r to the d. If you had such an example,  maybe you don't believe that such an example exists. If such an example did exist, its  orbit closure under this group would have very strong properties that sort of the, this  action detects, specifically detects this property of intersecting every convex set of volume  and since precisely because this group of affine transformations preserves the collection  of convex sets of volume one. Okay, so this was just a kind of an advertisement for this  ASLDR renormalization direction of study. And specifically, Mark Lofen-Strombergsen  asked two questions about this ASLDR dynamics. First question is, can we describe the invariant  measures which are supported on discrete subsets? So as I said, this collection contains all  closed subsets. The discrete subsets are some kind of a subset of this space and they want  to know which invariant measures there are. What can put on this collection? And a more  refined question, given such a measure, mu, and given a point in its support, is the following  true? So what is this statement? Maybe it's easier to parse this information when you  think about the SL2 case. So k lives in the group SLDR, so you might take it to be the  group of rotations. So this is just a circle. We're taking our set and we're rotating it  through a circle's worth and that's giving us a measure. And then we push the measure  by diagonal matrices. And so the circular action becomes a kind of a long ellipse. And  we're sampling the collection of closed subsets by performing this operation to our fixed  base point, y. And we want to know whether it samples mu fairly. It's a kind of an ergodic  theorem for a big group, the group SLDR, with these kinds of averages. And we want to know  whether there is such an ergodic theorem and for which specific points does it hold.  So this is definitely a difficult and technical problem in dynamics and it turns out to be  a crucial problem for the applications in statistical physics that Mark Loughlin-Strombergsson  are interested in. And so you might wonder if there are any such measures satisfying  these properties. And Mark Loughlin-Strombergsson showed that they are, that they are and they  are supported on, the discrete sets that they're supported on are cutting project sets. And  this, their construction relies on homogenous dynamics and specifically it relies on a very  famous body of work by Ratner from the 1990s describing the, doing the measure classification  for actions of groups generated by unipotence. Ratner's work is used twice. The first place  it's used is just to construct the measures. You're going to see that in a moment. And the  second place it's used, which I won't talk about today, is to prove this, understand  exactly this kind of equidistribution phenomenon.  So here is the construction of Mark Loughlin-Strombergsson. And since it's introduced by Mark Loughlin-Strombergsson,  it relies strongly on the work of Ratner. We're calling it, in this joint work, we're  calling it the Ratner Mark Loughlin-Strombergsson measures and it's going to be, the abbreviation  is going to be RMS. So these are what we call RMS measures. So what do you do? You fix the  dimensions of your cutting project construct. This is a measure supported on cutting project  sets. You fix the dimensions in the cutting project constructions. You fix the direct  somebody composition, physical space and internal space. You fix the window. So the only data  in the construction that I have not fixed here is the lattice that was used. The lattice  L or either the lattice or the grid. Now, with this data fixed, you consider the following  embedding of ASL-VR, affine maps of D-dimensional space into ASL-NR, affine maps of N-dimensional  space. Here's the algebra of the embedding. V is the translation, G is the linear transformation.  What you should think about geometrically is inside R to the D, sorry, inside R to the  N, you have physical space as a D-dimensional subspace. And you just let the affine map  move things in physical space without touching internal space at all, acting true for the  N internal space. So that's the meaning of this embedding. Once I've fixed this embedding,  I can consider the space of grids. This is a homogeneous space for this group ASL-NR.  And on the space of grids, suppose I could find any measure, which I'm going to call  U-bar, any probability measure, which was invariant under the ASL-D action, this group  via this embedding is acting on N-dimensional space because these are N by N affine transformations.  And so I take such a measure, U-bar, on the space of grids, such measures are associated  with certain groups H, which are intermediate between ASL-D and ASL-N. That's the important  work. That's the RACNOR measure classification theorem. And given such a measure, U-bar,  what you can do is you can project it to a measure on cutting project sets because you  have a map that says your grid, L is the grid, L is an element of the space, to the set lambda  LW, which is the cutting project set constructed from this data, including W, and from the grid  as in my first slide. Okay, so this is an important construction. I hope I've made it  clear what the construction is. If there are any questions, as I said, please stop me. So  this is the construction of Markliffe and Strombark Center. And I want to sort of give  you a teaser of the things we proved first and then make mention of conjecture. So the  first thing we prove is that any measure, which is ASL-D, invariant and ergodic, and  if suppose the measure gives full less to cutting project sets is one of the measures  I just described. There's no other way to put a measure which is affine and invariant  and ergodic and with probability one produces a cutting project set. And there's a technical  condition on the cutting project sets. They should be irreducible. Irreducibility for the  experts in the audience means it's a standard assumption in the topic of cutting project  construction. So irreducibility means that for almost every L, the projection onto internal  space is dense and the projection onto physical space is injected when restricted to L. And  so that's our first theorem. And I want to just mention a question. A question is, is  there any other interest in construction of a measure, not just on cutting project sets,  but more generally on Dillene sets. A Dillene set is a set that's uniformly discrete. So  the distance between any pair of points is at least some positive number and it's uniformly  dense. There are no arbitrarily large open patches without any points of your set. So  my conjecture is that any ASL-D, or invariant and ergodic measure, which assigns full measure  to Dillene sets, again, Dillene sets is a much wider class than irreducible cutting project  sets. Any such measure, well, I wrote has an untrivial joining with an RMS measure. Just  as a first approximation of the conjecture, just say, just ignore the words joining. Is  it true that any, can you, is it true that any such measure is actually the measure that  is described on the previous slide? Now, that's not exactly true. As I just stated it, what  you can do is you can take a measure that's constructed as in my previous slide and then  randomly, in a smart way, you can randomly add more points without destroying this uniform  discreteness property and without destroying the invariance and ergodicity under ASL-DR.  That can be done, but that's somehow cheating. In a sense, I started with a measure which  did come up according to my construction, to the construction I just outlined. Again,  so this discussion about the joining is a statement for which I have no counter example.  That's the conjecture. The only way to construct such measures is using the construction I  just outlined. Now, I don't have very strong evidence for this conjecture. The only evidence  that I have is that I tried several times using several different ideas to construct  such a measure, which is supported on this collection. You should remember that this  ASL-DR group is a big group. I wasn't able to. If you think, I'd be happy to be proved  wrong if you have a counter example, but I find it challenging to envision any counter  example to this conjecture. That was basically giving you an idea of what we care about and  why. I think this conjecture, if proved, would have very strong consequences for both problems  in mathematical physics, for the problem, the answer question that I discussed, and  other problems about discrete geometry. I don't think it's a curiosity. I think if  it's true, it's quite a strong result with a lot of consequences. Having said that, let  me move on to our results, which are not yet at all close to proving this conjecture, but  this was motivating the things I'm interested in. Now, I want to talk about our results in  more detail. For now on theorems, we'll be numbered. Let's just point that out. If a  theorem is numbered as it is over here, that means it's one of the theorems in the present  paper. The second theorem is a classification of the RMS measures, meaning Mark Lofen-Strombergsson  said that the measures could be described using Ratner's theorem, but they didn't actually  mail all of the data that comes up in Ratner's theorem. The data in Ratner's theorem involves  closed orbits for certain groups, and we give a list of the groups. That's the content of  the theorem too. Whenever we have an S of the R-Immvariant measure, we know by Ratner's  that it's H-Immvariant and support on a closed H orbit. Here is H. It's given by our theorem.  H has a structure of a semi-direct product. In the semi-direct product, the group of translations  is the entire group of translations in Rn. It can't be any smaller. The semi-simple part,  H prime, has a description in terms of restriction of scalars from a group defined over Q. These  are notions from the theory of algebraic groups, and I'm not planning to go into them in detail,  but what you should notice is that they're only two cases, and they give you strong conditions  on the relationships between D, which is the dimension of physical space, and N, which  is the dimension of the total space. One of the two cases, the symplectic case, is only  possible when D is two, when the dimension of space is two. Most, somehow, if the dimension  is larger than two, the only case that can arise is coming from SLK. I'll give you an  example now. Here are the three examples that can come up in the simplest non-trivial case,  which is when D is two, and is also two, and therefore N is four. The first case is, somehow,  the big measure, the measure of biggest support. The group H is the entire affine group of  four dimensions, SL4, semi-direct R4. The measure is the canonical ASL4R invariant measure on  the space of four-dimensional grid, which I'll denote by this M of YN. N is four here. That's  the first case. That's the big measure, so-called. There's an intermediate group, an intermediate  measure, which lives on block matrices. These are four-by-four matrices, and they're being  divided into two-by-two blocks. These are associated with real quadratic fields. I'm  giving you an explicit example, an explicit construction of the grid or the lattice that  will appear. If you want to use this group, you can't just use it. You need to find a suitable  lattice, and this is the lattice. It's very, very number theoretical. It involves this conjugation  in this quadratic field, and the Penrose vertex set can be written in exactly this form. In  the Penrose case, it's exactly case two. The third example possible in this dimension is  the symplectic case. The group H is the symplectic group, and the lattice is a symplectic lattice.  It has a basis, which is a symplectic basis of four-dimensional space. This is the sort  of an example of how the classification works in dimension two with N equals to four.  Now I'm going to give you a crash course on the Ziegels summation formula, which is what  is the Ziegels summation formula? The Ziegels summation formula is a certain formula that  was developed for studying lattices. What we're able to do in our project is to prove  analogous results for the measures on lattices for these RMS measures, which we've just introduced.  We have a bunch of formulas, which I want to describe. They're all analogs of work done,  all classical work done in the 40s and 50s. Here's the classical Ziegels summation formula.  Currently X of N here is the space of lattices of covalent one. Given a function on Rn, you  form a new function on the space of lattices whose value on a lattice L is the summation  over all lattice points, all non-zero lattice points of the value of the original function  f. For example, if f were the indicator of a ball, this would be just counting the number  of points in the intersection of the lattice with the ball. The Ziegels summation formula  from 1945 says that after normalizing by a constant c, we have this beautiful identity.  The integral of the original function f on Rn is the same as the integral of the transform  of f hat on the space of lattices. This is a quite important formula. It's used because  the measure on the space of lattices is quite hard to work with in practice. The easiest  way to work with it is through this identity. One of the beautiful things done by Markloff  and Stromberg's in 2014 is that they extended this formula replacing the measure on the  right, the measure on the space of lattices with an arbitrary RMS measure mu. Where the  transform instead of summing over all non-zero lattice points, you sum over all points in  your cutting project set.  As you see, I've written down the Markloff-Stromberg's result in this purple color. For now on  the results which are analogs of classical results, classical results will be written  in black and their analogs will be written in this purple color. This is the Markloff-Stromberg's  and analog of the Ziegels summation formula. A further classical summation formula is a  higher dimensional one. I've copied the original Ziegels summation formula here and I'm defining  a new higher dimensional analog. You fix another parameter p and then instead of summing your  function on vectors in your lattice, you sum over p-touples of p-different vectors in your  lattice and therefore the function f depends on p times n numbers which I think of as p  vectors in R to the m. This is the p-th Ziegels transform of your function f evaluated on  the lattice. There's an analogous formula where you recover the integral of this p-th  Ziegels summation from certain obvious measures on Rn to the p. For example, the Lebesgue measure  on Rn to the p or for example, measures supported on pairs of linearly dependent vectors or  p-touples with a certain kind of integer linear dependence and so on. There's a kind of an  explicit formula which is too complicated to write down which was proved by several people  and that's the analog of the Ziegels summation formula. But before stating the Ziegels summation  formula, one more thing you want to know is is this function in L1, is it integrable? Even  integrable just to get the conversation started. The first results in this direction are due to  Rogers and Schmidt from the 60s and strengthening is due to Eskin or Gullisonosis. I didn't have  room on the slide for their names so that's emm. Here's what they prove. If p is an integer  less than n and is the dimension then this transform is in L1. Sorry, yeah, then this  transform is in L1. A sharper result is if you take any q smaller than n, so q need no longer be  an integer, it could be any real number smaller than n and you look at the original  Ziegeltransforms written here, then its integrability exponent, it's in Lq. This is sharp. So the  largest, so for any q less strictly less than n, f hat belongs to Lq and as soon as q becomes n,  I didn't write that on the slide, as soon as q becomes n, f hat is no longer going to be in Ln.  That's more or less classical except for this new contribution from the 90s and we get the analog.  So we have the same two results for integrability balance of the transform in the case of cutting  project sets except the parameter n that appears here and here has to be replaced with this number  1 plus the q-rank of the group H prime that I had in the previous slide. So that's a  algebraic group theory invariant associated with this and as an interesting corollary what you can  see now is that if I fix the dimension d and I use different measures involving bigger and bigger n,  I can make this quantity larger and larger. In particular it means that for larger and larger  d, I can find analogs of this Zika summation formula with better and better integrability  bounds for the transform. That's an interesting fact that's not, can't be done for lattice.  We have more flexibility for cutting project sets. As I said earlier, once you have this  p-th transform, you can try to write down an identity for it and so this was done in the 50s  and 60s by Rogers, Schmidt and Andreve in separate work and as I said they have combinatorial formulas  for expressing this quantity and we have the analog for rms measures. We have kind of an infinite sum  which expresses these expressions and they also give us an expression for the higher moments  of the original Zika transform and the shape of the measures that appear is the flavor is given  by kind of a cutting project type construction for the big measure rather than for the discrete  measures. Another thing we prove is something called the Rogers second moment bound. So for  lattices in the 50s, Rogers proved the following fundamental fact. The this expression which is  the second moment of f-hapt, if you fix f and you think of this as a random variable depending on  the choice of the lattice lambda, then this is the second moment and Rogers bounded the second  moment in terms of the space into root of f which is as we saw earlier in the Zika transform. This  is actually the first moment, it's the expectation. So the second moment is bounded by a constant  times the expectation. That's a very important input to counting results and we prove the analog  theorem in the full context of an arbitrary rms measure.  And using this as an input, we're able to do something that Schmitt did in the 60s for lattices.  So what did Schmitt do for lattices? You start with any collection of sets which is which we call  an unbounded ordered family. So what this means is the sets are parameterized by positive real  numbers and when you have two, if you have t1 less than t2, then the corresponding sets are  contained in each other. So an example to think about is just a grand collection of balls,  Euclidean balls, but this could be a much more complicated example. You could you know  increase, take kind of a stripped very larger and larger or some kind of spiraling bodies or  whatever you want. It's just an unbounded ordered family of sets and all you require is that they're  each individually refining volume but their volume is increasing to infinity. And in this setup,  Schmitt discussed the typical growth of the point counting problem in these sets. So again,  just think of balls as a first approximation. So the left hand side is the number of points that  your lattice has with the large ball, say a ball of radius t. And Schmitt said the main term grows  at the volume which was known before and follows from the Ziegels summation formula. And Schmitt  established an error term which is on the order of the square root of the volume. So the quality of  this error term, the fact that we have a power savings, that's what I mean here by effective  counting. It's counting with an explicit bound on the error rate. This result of Schmitt from 1960  has not been improved. Even if you talk about Euclidean balls, not just this general, very general  setup of an unbounded ordered family, just take Euclidean balls and count the number of lattice  points in Euclidean ball. If you want to know the behavior of the error term for the counting problem  in a typical lattice, typically with respect to the natural measure on the space of lattices,  that's an open problem. And Schmitt's bound is the best bound known to date. There's a conjecture  of Goetze that this can be improved by reducing two by subtracting one over two d,  if I remember correctly, in the exponent. But that's an open problem and Schmitt's result is the best  possible. And we prove the pattern project analog of this result. So we prove an effective point  counting for the RMSH measures. We have the same exact error bound because we use the same technique.  And this involves these Rogers second moment bounds. With the second moment bounds, we're able to  look at the exact same asymptotic. And I wrote down here what I said earlier, it matches the best  known almost every bounce, even for the special case of lattices and Euclidean balls. Now, this is  an almost everywhere result. It's telling you the counting behavior of sets and balls for almost every  lattice or for almost every kind of project set. But it doesn't tell you what's going on for a  specific one. If you care about a specific one, say the Penrose example or other examples, then  there are other techniques. So for example, the Penrose example is better understood using  Perron-Ferbenius theory and the tiling, the substitution tiling machinery. This was done  in a paper of Solomon, Gar-Solomon. If you talk about the kind of project instructions, you can  get a better handle using Fourier techniques and diaphanthine approximation for certain explicit  lattices. I mean, the data and the kind of project construction should have certain  diaphanthine properties. And then you can get certain effective counts. They're not better than  this, but they are explicit. I mean, you have explicit examples of cutting project constructions.  So this is joint work that I did with Haynes and Kelly. And there's been other work using  Panamikri variant chronology on tiling spaces by Schmiedt and Trevino for cutting project sets  that have a certain dilation, self-similarity under dilations. Okay. Another thing that we are able  to do is to count patches. So I showed you earlier, I want to show the slide of the Penrose tiling,  what I mean by a patch. So a patch is you take your original point set, you centered it. If it  were instead of you move the center from point X to the origin, and then you look at the intersection  with the ball of radius R, and this is called the R patch centered at X. And obviously, if R is fixed,  you may see the same patch again and again at different Xs. And you might want to count the  appearances of this patch again in a large ball. And this counting problem had only been considered  up to the point of proving that there was a main term in the asymptotic. So there was a kind of a  prescribed order of growth. And this is sort of linked to the question of unique ergodicity. And  if you want to get an error term, in other words, if you want to get effective counting, it's kind  of like a form of quantitative unique ergodicity for these tiling spaces. And we're able to do that  for again for almost every almost every cutting project set where almost every is with respect to  any RMS measure. So fix an RMS measure, fix any window which satisfies a certain property of small  boundary in the sense of upper box dimension. Then you can find a power savings parameter,  which is this theta, such that for any patch, you have this order of growth of the catch of the  patch counting. There's a main term here. And then there's the error term. And the implicit constants  in the error term are going to depend on on the data. In other words, they depend on dealt on the  land on the kind of project set. They depend on the window. They depend on this dimension of the  window of the boundary window. And we get a power savings. And I just wanted to point out that this  result has no analog for that. It says simply because for analyze, it doesn't make sense to  cut patches. Every point has the same patch around it. There is no analog question. So this is the  first effective counting result for patches that we are aware of. Okay, I see that I managed to  put everything in my time. And that's it for me. Thanks for your attention and stay healthy.  Yeah, so thank you very much for the very interesting talk. So it's open for discussion. We  have some minutes time. So to start the discussion, just a small question from me. The constant  C lambda in the Schmitt like theorem. What is known about that? So they have an explicit formula.  You have an explicit. It's not our work. This is the main term constant was done by several people.  I don't remember the names right now. One of them being moody, but there was earlier work. And  it's expressed in terms of the co volume of the lattice L and in terms of the volume of the window.  It's just simple formula. Thanks. Yeah, Nikolai, our next speaker wants to ask a question. So please,  you can ask the question I can open. Yeah, you can. Just one question. So  I would like to ask if the results about classification of your measures  lead to certain mathematical corollaries of such a type. If a constant project set is  self similar, then something is algebraic.  Well, they should. And but it's in this kind of thing has been already done by  mayor. So may have this beautiful theory from the 60s, where he's proving that if you have a  what we call the kind of project set, may have caused it a model set. If you have a model set,  and it has the property that when you apply a dilation, the model set becomes the subset of  itself. This he calls this a similarity. Then the coefficient is algebraic integer and mayor  gives a lot of restrictions on exactly the algebraic integer that arises. It's part of his  beautiful, beautiful theory. We are not interested in these dilations. We're interested in  self maps, which come from the affine group. And in principle, the answer is yes. I mean,  it should be detectable. These kinds of things should be detectable from the algebraic data.  We haven't done it so far, but I think there's plenty of plenty of things of that type, just  which can be done. Yeah, are you satisfied, Nikolai? Yes. Yes. Okay. Thanks for that.  He can't help saying he's satisfied because he's the next speaker and I could ask him a question.  Yeah, of course.
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