 So thank you so much for inviting me to speak. I'm sorry in advance, it could be a few people in the audience who saw or heard me or my co-author Florian Richter talking about this. I still will be saying things behind the slides, which could be different things. And discussion could be different. So the second apology is to those who fully understand the topic. And maybe we'll find portion with a little bit boring. Sorry, I try to cater to wide audience. And so I will start. So we are interested in two actions. They would be semi-group actions, but these are nice semi-groups, concelative semi-groups, and plus and then times, the interaction of these two structures and integers is enormously interesting and fascinating. And what we propose is to look at new kind of ergodic theorems, which in a sort intertwine these two actions. We will see these two semi-groups. We'll see how it goes. So the very first page, I just want to set up some notation. Again, apologies. That's totally well known. But since I'm in US, so my n, the natural number starts with 1. And we all know what the primes. So one of the heroes is omega n, large omega n, the number of prime factors of integer n. And then maybe occasionally we'll see small omega n. So in capital omega, when we count them as multiplicity and little omega n doesn't count them as multiplicity, and there are people here who know these things much better than me, it seems that on many occasions, the synthetical properties and all kinds of averaging results along capital omega and small omega are very similar. Now, of course, there is also Liouville function related to omega n. Here is definition. And there is, of course, Möbius function. Möbius function is especially attractive for a variety of reasons. It connects to square free numbers, which are my favorite. And they have a very nice natural density. In any case, I will keep going. And now we can discuss some classical results. These are well known to many of you, some equivalent forms of prime number theorem. And so the first one just tells us that those n for which omega is even, I call them sometimes multiplicatively even numbers. And those for which omega is odd, they have additive, natural density one half. And this is very nice and pleasing fact, but it is deep or deep enough. It's equivalent to PNT. And I was able to trace the authorship to Von Mangold and Landau. If there are people here who know more of history of these statements, I would be very, very glad to know about. So again, the fact that lambda n averages to zero is yet another form of PNT. And that mu n, this is the previous sum, and now we skip the square free numbers. So all these results are classical versions and forms of PNT. And we will try to see if they can be embedded as special cases of some naturally enough, dynamically naturally enough, ergodic theorems. So, but let us stay a little bit on results about distribution of omega, capital omega in different classes of numbers. So natural generalization of the first bullet is so-called Piliy Selberg theorem, which says that the omega n is equidistributed in these modular sets. And with a correct distribution, so to say. And then there is a little bit more modern results due to Erdos and Delange, that if alpha is irrational, then you again have a sort of a similar result. So I hear a formulated in a shape close to just the definition of uniform distribution. And the interpretation for us is that if you take these sets, alpha m plus beta, they are sometimes called BT sequences. And they're popular in some quarters. And so they have natural distribution in these sets. The natural density of omega of n for each omega when belongs to such alpha m plus beta set is one over all. So if you believe in a very good equidistribution of properties of omega, that's all guessable. Or at least after you see a few of such results, you start thinking that they're very nice, nicely distributed. So let us move now to our body capricious. So just to set up terminology, additive topological system is just the pair. You take a, you have a compact metric space and you have a homeomorphism or continuous map. Now it's not really strictly speaking the pair X team, but the pair X and all iterations of team, which is a topological dynamical system. But if you have T, you can take iterations of course. Now, and there are many classical such systems which have extremely good properties also from measure theoretical point of view. And so let's discuss this. So we start with pure topology, topological compact space and a continuous map. And then there is classical result by Bogolubov-Krylov, which says that any topological system has a team variant probability measure on the Borel sigma algebra. Sometimes it may be just the fixed point, which is not terribly interesting, but suppose you have an irrational rotation on unit circle, then it is all familiar hard measure. And we always try to normalize this measure. So I talk here about finite measures. It's totally separate and pretty interesting for ergodic people question, what about sigma finite measures? We will not touch it at all. So our measures will be always normalized. And so if there is only one such team variant measure, we call the system uniquely ergodic. Now why ergodic? Because if system is, if this measure is unique, then system is automatically ergodic with respect to transformation team. Again, the example I gave you already, the translations, irrational translations on torus are uniquely ergodic. That's actually easier exercise. You can try to do it two ways. One way, think of alpha, irrational, then N alpha is dense and then invoke uniqueness of hard measure. And you may want to do it other ways. So I will leave it to that. Now there exists another interesting example of systems maybe will, here it is, skew products. So look at this one, X, Y going to X plus alpha, Y plus X. It's called skew product in ergodic theory. They are very nice systems related to uniform distribution along polynomials. We will maybe refer to them later, but before we go to some additional formulations or rotations I already mentioned, but the simple system, cyclic permutation, any finite set is also, and this is unique for actually. All right, are there any questions so far, especially about ergodic part? So it seems that there are no questions, I will keep going. So here is a classical fact. So we all know that ergodic theorems are about, and let me be philosophical, they're about equality of two things, time average and space average. That's what is ergodic principle and it's very prominent in number theory. Say uniform distribution is nothing as an instance of this ergodic principle. So on the left, you usually have Cesaro averages, this is time average. You may think of your system T corresponding to measurements every second, okay? This is time average. And on the right, you have a space average. If my space would have measure, which is not one, I would have to divide by measure of the space, but my measure is normalized. So the right-hand side is just space average. So ergodic theory is about equality of time and space averages. Now, if your measure is unique, then you expect this to happen not only for almost every point. If you have an ergodic system, there's classical point-wise ergodic theorem tells you that for almost every point, the orbit is uniformly distributed. Here, our life is way better. First of all, it's for every x. And second, which I didn't mention here, but these averages can be taking along any sequence of increasing intervals, increasing in size. Indeed, for those who know what is amenability, you can take any fielner sequences and it will be still the same. There is one, though restriction, unlike the classical ergodic theorem, which is valid in LP spaces and at one sake, here we want to restrict ourselves to topologically sound function. So here I wrote continuous, but any Riemann integrable will be fine, but not more general, really, okay? So here is our theorem, one of the few that we have, but this is a simpler formulation. So take now uniquely ergodic system. And now it's a funny mix of two things. Instead of taking t to power n, like we had it here, right? So that would be natural for ergodic theorem as actions of additive integers. Here I take it to power omega, same omega that we introduced on the first page. And it turns out that if system is uniquely ergodic, then you still have correct limit. And this result has multiple applications to number t, which we will list now. And it's really a beginning or a quest for something even more maybe dramatic. But in any case, this result itself has a simple formulation and let's see what you can get just from this result. And by the way, to think again ergodically, what it means that you take a point x and you consider omega orbit. So to say you just consider power is a long omega one and they uniformly distributed. That's what is written just here. Okay, so what are the applications? Take the simplest possible system, rotation on two points. x going to x plus one, mod two. And take a simple function f of zero is one, f of one is zero. And then it takes maybe 20 seconds to think a little bit but you will see that the left-hand side then becomes density of the set of those n for which omega one is even. And this is one half because integral of this, don't forget, my measure is normalized. So integral of this function is one half. So this is one of the formulations. I think I can jump here. This is the very first bullet on page two. So we got it by applying this ergodic theorem to this simple rotation on two points. Now let's go, by the way, I call it Pillai-Selberg. Maybe I should call it, I think I confused a little bit between, I know Pillai-Selberg is okay. Later I will be talking about Erdisch-Pillai. That will be a confusion, Pillai-Selberg is fine. So let's go back to this statement. So now I take a system on M point, it's rotation mode M, and very similarly to previous example, you will get equidistribution of omega in those modularly defined sets. Now let's go to Erdisch-The-Launch. Actually, surprisingly, the proofs of Erdisch-The-Launch, Erdisch actually, I think only mentions it and says that proof is not totally trivial and the lunch gives a proof. And here I have to make, by the way, a apologetic statement. We are getting these nice results relatively easily because we don't care about or can't maybe handle them well, they're bound, we don't deal with serious number theoretical estimates. The methods of classical number theory, they involve the estimate sometimes as part of the process of the proof. So because our approach is softer, we cannot say anything new about any estimates, but we can show maybe new results and get the old result in relatively unified fashion. So Erdisch-The-Launch is also corollary of this theorem A and to see it, you take again this function and you will consider rotation by irrational alpha. You substitute this continuous function F into my system, into my formula here in theorem A and you will get a wild criterion form of uniform distribution. So we get, sorry, Erdisch-The-Launch also is a specialty. Now, attention, we already got a few pretty nice corollaries by considering pretty simple, I would even say primitive dynamical system. So let's take one step up and what happens if I would consider systems like this one, or maybe it's more general forms, that's the skew product can be defined on any finite dimensional torus. This is actually a special case of so-called NIL system. And NIL systems are known to be uniquely ergodic on their minimal components. So in principle, we can expect additional interesting applications because there is a good supply of natural uniquely ergodic systems of algebraic origin. So here's one of such applications and that's where I'm confusing names. So I'm not sure I should call it Erdisch-Pilai, maybe I should call it Erdisch-The-Launch, but sorry about that. But here is the formulation. So take a polynomial, standard polynomial of that kind to which while the theorem on uniform distribution of polynomials applies. And if you substitute omega inside, it is uniformly distributed. And I'm not sure, but it seems that this polynomial version is actually a new result. I could not find it in classical literature and it's pretty nice. So substituting omega of N instead of N keeps uniform distribution. This is part of something bigger. So I will talk about this later, but let's remember this. The fact that Q of N is uniformly distributed, this is classical result of Y. And you can also prove it dynamically if you use the skew products. This was by the way observed already in thesis of Hillel-Fürstenberg. And we are using this, namely the idea of applying the dynamics. Namely, I can generate such a polynomial by using skew products. And then because skew products, if ergodic are uniquely ergodic. And they are ergodic because we assumed that at least one of the coefficients from C1 to CK is irrational. So then again, it's just application of our theorem A. Now what is, but we don't have to stop with skew products. So there is a even bigger class of interesting uniquely ergodic system. These are translations on many faults. So I didn't define it here. Let me in few words explain how one can think about this. Take three by three matrices, which are upper diagonal ones on the diagonal and the X, Y that's a, suppose it is three by three. So matrices three by three, ones on diagonal, zeros below diagonal and above the diagonal real numbers. And now there is a subgroup, discrete subgroup of the same kind when the entries above the diagonal are integers. So factor out, it will be sort of similar to how we create three-dimensional torus by taking R3 over Z3. And this would be an example of a nil system. Now, if you apply theorem A to nil systems, then invoke a result which I have with Sasha Lehmann, then you can get even more impressive uniform distribution results. So let me first of all define what was generalized polynomial. And I would like to make some propaganda for generalized polynomials because they turned out eventually to be indeed very interestingly connecting many sub areas of combinatorics, number theory and ergodic theory, but they start to, the definition is more or less natural. You just take regular polynomials like QoN and you allow one more operation taking integer part. Well, if you end the rest of the rhythmical operations, if you allow integer part, you should equally allow say fractional part because it's expressible through integer part. So in other words, not so formal definition of generalized polynomials is take a regular polynomial and sprinkle brackets inside, just sprinkle them properly. So there is always two in your right bracket, left bracket corresponds and so on. This will be generalized polynomial. So as a very special case, for example, this expression is a generalized polynomial. Now I already put omega in, but suppose it would be just n alpha bracket beta. This would be a generalized polynomial. Pay attention that in this theorem, the first bullet on this page, they will already some primitive generalized polynomials because the results are both mod one. When you take mod one, this polynomial mod one is already a generalized polynomial because you used fractional part. The point is that you can use fractional or integer part as many times as you want. And now we have the following result. So before I will come back to its formulation here in the middle, I just want to tell you that we have a theorem with Sasha Lehmann, which says that any bounded generalized polynomial comes a very dynamical way. It is just f of t and x, where t is a uniquely ergodic nil translation. And f is not necessarily continuous, but nice enough, say Riemann integrable function. It's actually a little bit better than that than general Riemann integrable, but that's all which we need. And so just because of this, and so then if I take generalized polynomial, for example, n alpha beta and put instead of n omega, it will be n alpha bracket n beta. And they put omega instead of n, I will get things of this kind. So here is the more general result. So take any generalized polynomial. I'm using q for polynomial, so this q is written as if it is rational numbers. No, it's just a polynomial q, okay? And take any generalized polynomial and assume that q of n is uniformly distributed, then q of omega n also is. I intentionally formulate it in this form because I think there is enormously interesting principle behind it. So let me spend a minute on this. So this adds to collection of theorems that have the following feature. You have some uniform distribution result involving parameter n and you put instead of n n square and it's still uniformly distributed. Or to have even bigger picture, suppose you take an ergodic theorem, which goes along ends and then instead of n I will put n square. And then it's still an ergodic theorem with almost the same conditions which I needed for it to give correct limit. Jean Bourguin proved that almost everywhere even is true in LP spaces for p bigger than one. That's a little bit too technical, but the rule of thumb, which we observe in ergodic theory and in combinatorics is that you can replace n by polynomials of n and things still remain true. This actually applies to another theorem which Sasha Lehmann and I have, it's called polynomial semi-ready theorem. So semi-ready theorem tells you that any set of positive density contains arbitrarily long arithmetic progressions what Sasha Lehmann and I proved that it contains arbitrarily long polynomial progressions. You allow jumps not just n to n, 3n, any polynomials which say have zero at zero to avoid local obstructions. So we have yet another instance of this very, very interesting principle. Replace n by n square or replace n in this case by omega n and results still remain true. So to better understand what's going on, I think it's very interesting direction. For thinking. Okay, so this is an instance of classical results remaining true. You see here, sort of double substitution. First of all, we had polynomial uniform distribution. That's why that itself is very interesting. And then you put instead of n omega n, it's still true. So that's something, some curious principle which needs more development. All right, now let's go back to P&T statements. You see all the statements, they already statements on average and they can already interpret it their body quake because these functions take values plus minus one. So evenly that you can think that the integral if it's involved there should be zero. So the time average is the same as space average. That's not too far fetched but to make it a little bit even more visible take any uniquely ergodic system and let us take this pattern. You take the ergodic limits along ends which are square free like in this version of P&T. If you use this with some improvements in now theorem A, you'll get this result. This result is really open to nice interpretation. So let's look at it. Of course, six over pi square appears here because of the density of square free numbers. That's if you will integral of the characteristic function of square free numbers. And you can think that in the left-hand side you would multiply by indicator function of square free numbers then it will help you to see that summation is only over square free numbers. So in the left you have product of two functions and on the right you have product of the integrals. So that's a manifestation of ergodicity in new convenient shape name it's independence. Okay, so we'll come back to independence and try to analyze it a little bit more carefully but I suggest to see this result as independence. I repeat why independence because on the left if you multiply left-hand side by indicator function of square free numbers it's product of two functions and the limit is product of their integrals. In generalized sense I think of six over pi square as integral of characteristic function of square free numbers. So here is a guessable if you start believing in this you can guess this result, right? So take now summation along k free numbers and the limit should be if there is any justice and in this case there is justice it will be one over zeta of k times integral over. Okay, so it's time now to introduce multiplicative dynamical systems but we will see soon that these results about independence they also manifestation of some very interesting principle which tells us something about independence of actions of n plus and n times and that's the goal of my talk. So one observation before I will make general definition. So so far we were considering t to power omega over. This is in our theorem we can keep jumping. So theorem A, oops, that's not this theorem A not this theorem A. So if you consider t to power omega over n and because of properties of omega over n then just let me move. Yes, then you will see that this is an example of action of multiplicative integers, right? So because of properties of omega t to power omega over n times m is t to power omega over n times t to power omega over n. So why don't we define it now generally and try to maybe generalize. So let me remember I'm going to page seven. If I go back to page three, indeed four. So maybe I can try to make a more general expression on the left, which will somehow involve both additive and multiplicative actions. And then on the right I will expect some independence manifested by product of integrals. That's what we are planning to do right now. So definition. So we know already what is additive topological system just a homeomorphism or continuous map and its iterations acting on a compact space. And now we can do the same with a multiplicative integer. So it will be a pair, ys, y is compact metric space and s is an action of multiplicative integer. So you note it was s sub n. And the postulated property is that s sub n times m is s n times s n. That's the action of multiplicative integers. And we had already this example. So it gave us a lot of applications. Now we can define formally and conveniently multiplicative rotations, so to say on torus by taking any multiplicative function, bo m, and defining action s sub n of z to be bo n times z. z is in s1 in the unit sort. And this will be already an action. And of course then we want to make y to be compact space. So I take just this orbit and take a closure. So there are many ways of defining multiplicative topological linear systems. Here's one more just for you to realize that there are many of them. Take any subset of integers and assume that it is a non-trivial from the point of your multiplicative structure. You can take it's orbital closure, so to say. You take the space of all sequences. And any subset is a zero one sequence, right? Any subset is identifiable with its indicator function. So subsets in n are the same as zero one sequences. The space of zero one sequences is a nice compact disconnected space. There you can take orbits, orbits under some action either under action of shift that's z plus action or you can multiply the coordinate by m. So it was z sub n or x sub n and becomes x sub nm. And you can take these sequences in this symbolic space of all sequences and take their orbital closures and you will get a lot, a lot of multiplicative systems. It's meaningful when you want to deal with generalized notions of normality. So it's totally not my topic today. So back to multiplicative systems. There is a version of Bogolubov-Krylov. So Bogolubov-Krylov theorem still is valid for any amenable group or nice amenable semi-group and n times is amenable, nice semi-group. So you have always n invariant measure, invariant now with respect to action of sn. If such a measure is unique, we'll call it uniquely. So then the question is, is the theorem A that we formulated at the beginning and applied a few times, is it a special case of a more general theorem dealing with multiplicative systems? Because t to power omega is a multiplicative system. So now I will have to ask you to trust me that these somewhat restricting definitions are not too restricting. And it's a special discussion how to relax it or if it will help. But let us assume for now that my system is, remember, so far we dealt with single t, t to power omega. I'm ready now to allow finitely many such t's. So my system is finitely generated if the s, the generators of sn and these are the generators of my action are sp, for p being prime, right? Everything can be made as product of prime. So I want this set of primes which I involved in generating sn to be fine. That's already much more than single t, but it's still pretty manageable. And second assumption, which I don't want even to start defining, it's a little bit too technical, but trust me, it's not terribly bad. It's a unique ergodicity and a somewhat strengthened form. So we call it strongly uniquely ergod. And then we have the following analog of theorem A, but in this more general shape with expectation, it will give us additional applications. And so the theorem is that if you take a finitely generated multiplicative dynamical system, which is strongly uniquely ergodly, then you have the same kind of result. And now let us see, so this is theorem B and the corollaries of theorem B are already involved. First of all, more general form of some, sorry, question there. Integral of F is the integral of G. Oh, very good point. Thank you, G equals F in the statement. You have a sharp eye, yes. F equals G, I forgot to tell you. So you should assume also, let take also F and G and assume that F equals G. I'm joking, but yes, F equals G. Thank you, thank you so much. Let me for now skip this corollary because I'm worried that I will not have time to discuss something more fundamental that I want. And now it is a attempt to take this independence, influence to look at the Peter Sarnak's conjecture. So let us introduce again some definitions. So you take two bounded arithmetic functions. So A, O, N, N, maybe complex value. So you take A, O, N times B, O, N and sum it up. And you compare it with a product of averages of A and B separately. If this limit is zero, then I hope you agree that we can call them independent because each, the limit of each of these averages can be generalized sense to be thought of as integral of A, O, N. And so the products, the scalar products go to correct limit. So it's independence indeed. You can also interpret it as orthogonality if you want, if those, at least one of those averages goes to zero. And what we want to do, and it is in line with what we discussed so far, we are interested in those cases when this formula star holds for two sequences coming from two different worlds. One comes from ADT world and one comes from multiplicative world. Or because we will be considering orbits, one comes from ADT topological system and one comes from multiplicative topological system. The examples which I showed to you so far indicate that maybe there is some, let's call it disjointness, phenomenal. So what is formal definition of disjoint here? And I apologize, disjoint is overloaded world. And we just couldn't find something better to describe what we want, sorry about that. So you take an additive system and you take a multiplicative system. Remember XT is additive, you consider T together with all of its iterates and the S contains a sub ends which behave multiplicative. So we call these two systems disjoint and it's pretty strong condition. If for every point in X, every point in Y, every function F in C of X, and this time I want G indeed not to be equal F, and G a function in C of Y, if I want all the situation when I define A of N to be orbit of X, additive orbit as you see and B to be orbit of Y in space Y, I want A of N and B of N to satisfy formula star. So in this case, I will think of these two systems as disjoint. And let's look at examples. Maybe the earliest example of side disjointness is a classical result of Davenport. So he proved this, let's call it orthogonality or liberal function with E to power two pi in alpha. But we can now interpret it as an instance of this disjointness, how? So multiplicative rotation on two points, what is sitting behind this is the following statement that multiplicative rotation on two points is disjoint from additive rotation X going to X plus L. Okay, now let's go to WC. So he has this beautiful result for any irrational alpha and then completely multiplicative function B, you have this, again, let's call it orthogonality. You see, I'm not sure what was motivation of Davenport and WC, but for me as an robotic person, it's so easy to see this indeed as instance of independence, indeed the instance of what we call this foreign disjointness. So reformulation of result of WC is that if you take, again, a rotation, but this time you compare it, this rotation on torus, you compare or put against multiplicative rotation, Z goes to B O N times E, that was a simple example of multiplicative system that I described above. And so we know now that these two systems are disjoint because of result of WC. Of course, if you would prove a theorem of general enough kind, then this will become a corollary. That's one of the things that we are doing with the foreign. Now let us talk about Peter Sarnak's conjecture, which was indeed the motivation for this whole business. So Peter has to be praised for giving us a lot of work in ergonomics because this Sarnak's conjecture has so many facets of robotic theoretical nature, which are fascinating. So here is one of formulations of Sarnak's conjecture. You take lambda is our Liouville function, which is very random as we know. By the way, let me stress random additively, okay? And they take a deterministic function, which I will talk a little bit more in a moment about, and then they should be orthogonal. This is orthogonality, or if you want independence, or if you want disjointness, okay? And what is behind this? Well, if function is deterministic, it comes from systems having topological entropy. I will not start introducing topological entropy, but this is a very basic today notion in ergodic theory. And the topological entropy being zero corresponds to the terminus. That's the philosophy in ergodic theory. Systems which have zero entropy can be otherwise pretty or would be chaotic. For example, it can be weakly mixing or strongly mixing. For those who know what the strong mixing, but it still may have zero entropy. And that's what is needed for determinism, so to say. So another formulation of Sarnak's conjecture is that the multiplicative rotation on two points, that's our Liouville, is disjoint from any zero entropy additive, topological dynamical system. And I hope after all I said so far, it's tempting now to try to replace lambda over N. You see, A of N is already of the kind we discussed here, right? A of N is already of this kind. If I can replace lambda by something more general, then it will be a dynamical way of thinking about general framework for Sarnak's conjecture. And so let us try to do it. So here is a heuristic principle. It's intentionally called heuristic because it's a philosophical state. And so let me state it. So you have a zero entropy additive topological system and you have a low complexity, multiplicative topological system. Pay attention why I say low complexity because multiplicatively lambda over N is very low complexity, okay? So since I want to think now about lambda over N multiplicatively and generalize this. I don't care about low complexity. Now for any amenable group action or semi group action you can define entropy. So it would be tempting to say for any zero entropy additively and zero entropy multiplicatively you have these joints. That's actually not true. That's why I say low complexity because that's very interesting challenging direction of thinking so what should it be really? What low complexity means? But here are some remarks showing to you that low complexity cannot be just zero entropy. So here is a simple example. So and indeed perhaps the most natural example of multiplicative semi group action you take instead of our familiar two X mode one and it's iterations you take N X mode one. It's easy to see that N X mode one forms a semi group action of N times. But if you take them and by the way for all kinds of scaling reasons this is an action with zero entropy. The reason is that for any fixed N it has positive entropy. I don't want to go into this but this is an action of zero entropy as action of N times but it's not good for us because you can create from this action deterministic sequences of this form and you will kill this principle. So you also want to avoid local obstructions and that's why you have to introduce notion of a periodicity. And so here is a definition of a periodicity. You want independence from any periodic sequence. I hope it's convincing definition and you want similarly to define a periodicity on multiplicative level. Now here is a conjecture. By the way, I hope I am doing well. Time-wise I have only one more page after page 12, right? So I'm okay. Okay, good. So here's our conjecture. So you take additive topological dynamical system of zero entropy. You take YS to be finite. Now it's not principle it's conjecture. So we hope it's actually true and probable so to say. So you take now finitely generated multiplicative topological system and if either of them is a periodic then they are disjoint. Now, Sarnak's conjecture corresponds to special case of conjecture one when Y is just multiplicative rotation on two points, that's the Louisville function. And so here is a, I say aesthetically appealing version maybe only for robotic people, sorry about that. But here is a version of our conjecture. You take a uniquely ergodic additive system and finitely generated and strongly uniquely ergodic multiplicative system. If either of these dynamical systems is a periodic. Remember before I was defining a periodicity for sequences I can define it for systems. Then you have this indeed to me looking a statical relation. So averages of F of T and X times G of S sub and X tend to independent. Some of the results that I showed before were exactly of this kind and you know to prove. And an instance of this theorem that you know to prove is that if XT is a nil system. So nil systems by the way known to be of zero entropy it has a relatively easy reason. Nil systems are so-called distal systems. Systems are distal if no two points go too close along the orbits. And it was an old and relatively simple result of Bill Perry which says that any distal system has zero entropy. And so nil systems fit the profile and for them we can prove. And one of the application is here which is a general form. So you take a basic which almost periodic function. So let me remind you that functions are basic which almost periodic. They are defined similar to war almost periodicity but the metric is different. It is Chisaro averaging on that. Let me not go into this but these are natural functions when you deal with integers. So almost periodic functions are more natural when you deal with continuous mathematics but when you deal with discrete mathematics basically which seems to be more appropriate. And so this is of course an example of disjointness and the special case will be what I already discussed about. Namely this theorem which can be now viewed as disjointness of characteristic function of square free numbers. It is Bezikovic by the way. It is Bezikovic. By the way I wrote here Bezikovic but it is of course Bezikovic. And so Bezikovic because you get to square free numbers by manipulating inclusion exclusion with infinite arithmetic progressions, okay? And so to finish for today I will formulate one more curious uniform distribution result which seems to be quite new. Take just any two polynomials. Let's call them wild type polynomials. Those polynomials for which you have uniform distribution. So you keep P of n here and in Q you substitute capital omega. And this is still a uniformly distributed sequence in two dimensional poles. Actually, I think we know to prove the following which is even more impressive in a way. Substitute instead of the first n in this P of n, substitute little omega. This pair will be still uniformly distributed. So P of little omega comma Q of large omega as a two dimensional sequence it's distributed in two dimensional terms. And at this point I should stop I guess because I talked for about 50 minutes but I will be very glad if there are any questions or discuss. Thank you for that.