 Thank you very much for the invitation and for coming around. I want to talk about density of rational points close to manifolds. And I'm going to start with the question of rational points on algebraic varieties. That's where I came into this field from this point of view, in particular about the dimension gross conjecture and a brief sort of account of what this is about. Then I want to talk to this more general question of looking at a manifold and rational points close to the manifold and explain how this has shed light on the rational and the dimension gross conjecture in recent years. And in the last part, I will then outline applications of also some recent work, including collaborators Ratschler-Schribastava and Niklas Technow-Meinz to defend an approximation. So let's start at the very beginning. If I take a projective variety, say, given by our homogeneous polynomials with individual coefficients, let's start over the rational numbers Q for now, then I obtain a projective variety, say, V. So this is zero state of my R, homogeneous polynomials. Then some very basic questions that you may want to ask about your variety V is, for example, about the existence of rational points. Of course, a very deep and vast topic, which I'm not going to touch upon, but another question could be if there's maybe infinity maybe points, how about a distribution of rational points on V? In terms of distribution, if you have infinitely many points, you may want to count them up to a certain height. So let's first define some height function. Let's say, again, V is our projective variety. Let's assume for now it's irreducible under a certain degree, say, D. If you have a point in projective N space, say, X in homogeneous coordinates X0 to Xn, then we can always find a representative where all the coordinates are integers and co-prime. There's exactly two such representatives. On this plus or minus sign, you can change the whole vector. Then we define a height. It's a max of the absolute values of the coefficients. So this is well-defined. And this gives you what we call an naive height on there. So given we have the naive height, you may then ask the question, if I look at my variety, V, and the rational points, and then I can ask how can you maybe count the number of rational points of height bounded above, say, B, where B is some parameter which you may envision going to infinity. In particular, we're going to talk about a batch number of upper bounds for this function. A first trivial upper bound is given by the dimension B to the dimension plus one. You can prove that by induction, basically by slicing, if you look at the affine cone. And we're going to look now at one concrete example here. So this is, again, the definition of the height function. If you have a point in Pn, the height is given as the max of the absolute values of the coordinates if they are given by a set of co-prime integers. If we then count rational points on the variety Pn itself, then we want to count all such two both x0 to xn, that a GCD is equal to one, coordinates bounded by B for all i. And we have to divide by a factor of half because there's exactly two representatives, the vector, and minus its vector. They give you the same point in projective space. So this would be a concrete counting problem. And you already see, if you for now ignore the GCD for a moment, then for each of the xi's, we have roughly B to B plus one choices. So you should have something of odd of magnitude B to the power n plus one. It's actually true that GCD only changes the leading constant, the number of rational points of naive height bounded by B grows like B to the n plus one. This matches, again, our trivial bound with the dimension plus one. So the dimension of Pn is a projective variety. It's n, so we get it B to the dimension plus one back. But this is not a hard theorem. You can also do such things over a number of fields and proof asymptotics, where the shadow will. But for us, this is now enough. And I want to reformulate it now for a moment into a linear subspace. Well, it's basically the same thing. I just embed in a different way. If say V is the linear subspace in Pn given by one linear equation, it's not the general. Then the dimension of this linear subspace is n minus one. And by the same arguments, we get up to a constant B to the n points. So think about choosing x1 to xn freely, and then x0 is uniquely determined. Linear subspaces tend to have many rational points. And you may now ask the question, if we assume that we do look at irreducible varieties, which are not a linear subspace, can we maybe get some non-trivial, upper bound some improvements on that. Some improvements may be so wrong about some stronger upper bound. If you move from linear hypersurface to quadratic hypersurface, as all the AIs are non-zero, say a full rank might work here, then one can count rational points of bounded height. And you will obtain that there is constant times B to the n minus one points on there. I assume n is at least 4 for safety, because otherwise there may be at some point a logarithm popping up. This can, for example, be done by things like the circle method. And so heuristic of how to arrive, there could be that you say the xi's, they all have size at most B. They're all integers. So it has something like B to the n plus one choice is for your variables xi. And this is a quadratic equation. So if the values were all, say, randomly distributed in the image where they could land, which is an interval of lengths around B squared, because it may be a typical value of this quadratic polynomial but stick in variables of size B, maybe size B squared, then if they were distributed equally in this length of size B squared, it would give me the probability of B to the minus two, that this is zero, this heuristic would lead to the asymptotic B to the n minus one. You can now do the same thing for larger degrees. So if we say take here degree D in the diagonal hypersurface of degree D, you can look at the same equation. And by the heuristic, I would now still have B to the n plus one choice is for the coordinates but a degree D equation. So say the probability of this. So again, it will distribute it would be B to the minus D for one such sample choice to be a solution to the equation. One can actually prove such things for degree D hypersurface. This is true of roughly n is of size D squared, which uses a resolution of nucleotus mean value theorem. You can then we learn Gussner, general case of a really previously on a degree three and also what's general later on. I don't go and talk about the fine details here, just want to sort of point out that what it seems like if you look at this question from a naive point, if you had a higher degree, what are the last points you'd expect here? This is just an example, but one can phrase much more generally in terms of mining's conjecture for rational points and varieties. If you read here this n plus one minus D and you think from an algebraic geometry point of view, you may say like this n plus one minus D is if you look at a canonical sheath of the endless of minus n minus one, the canonical sheath of the hypersurface is O of minus n minus one plus D. So the anti-canonical sheath is exactly given by O of n plus one minus D. And this is exactly sort of this matching exponent. In other words, if you would use an anti-canonical embedding of this variety, you would somehow normalize this exponent away. Sort of that is exponent is somehow related into looking at an anti-canonical embedding that's part of what is formulated in mining's conjecture for rational points and projective varieties of a number of fields, but we're not going in this direction. Despite it seems that higher the degree is, the less points it has, this is all conditional on this varieties being nice enough. If we are given a variety and you don't know anything about it, maybe you don't know if it's smooth or anything, the worst things can happen. So this is a very simple example, but just to illustrate a point, if you don't make any assumption, say look at this example, a hypersurface again in PAN, take some polynomials, should be homogeneous of the same degree of zero F one, then look at equation X zero F zero minus X one F one zero, that contains many rational points because I could set X zero and X one equal to zero. So we produce a high dimensional rational subspace on there. If you choose X zero and X one equal to zero, other coordinates are free. So you should expect B to the N minus points on there, which is again B to the dimension. And you can make the degree as large as you like if you have this sort of linear subspace container, that kind of singularity which is reflected and you can't do better. Dimension close conjecture gives indeed an upper bound which works or which tries to cover all projective varieties or Q of a given degree at least two, no matter how singular they are. And the conjecture is that indeed, this example that we've seen on this singular thing and the same that thing that we have seen for the boardwalk is the worst situation that can happen. So the conjecture is that if you have an irreducible projective variety of degree at least two, then the number of rational points and there is bounded by B to the dimension plus epsilon. So that matches this example because N minus one is the dimension of your hypersurface and also the one for a quadrack. It's called weak dimension close conjecture. And you could also conjecture stronger things and also in certain situations, stronger results are known. There's for example, a plus epsilon, you could debate when does it actually necessary or when it should maybe disappear. I put here a V in the absolute, in the constant. You could also here ask maybe there's some uniformity in terms of the degree or dimension of the variety. Both of these, there's a meaningful, yeah, there's work results in this direction. So indeed, sharper versions exist in some situations in the weak versions already solved in the work of Browning, East Brown and Saltberg are using their determinant method. So that's a very, say, a arithmetic approach where you try to locate your rational points in some auxiliary sub varieties and then try to intersect those sort of by pursuit type styles so we'll just get our bounds in the end. And just let me mention here along the way, we see very nice work of Astrid, Klugeus, and Lien and Dittmann who managed the least degree, at least five to remove the epsilon and get even here a polynomial dependence in the degree. So not really a uniform result but even a very explicit one. So this was an introduction of one side where I think I got interested in this area. Instead of counting points on a algebraic right, you could also try to count them close to a variety or more generally you could take any manifold I started now just with a board we could just not define over Q. You may now ask here for integer points close to the rational points which are close to it. This example by open minds conjecture where we'd get arbitrarily close to this quadric. But so imagine like instead of you're lying now in the thing I want to have points close to being somewhat just in a neighborhood. We can formulate this as follows. If you have some bounded set manifolds here of dimension little m that's gonna now stated dimension for the rest of the talk here. Then we define a counting function which comes with two parameters. First capital Q and delta. Q measures the complexity of your rational point the size of the largest denominator. If I look at a rational point and say parameterizes with P over Q where P is an integer vector and Q is a natural number of size up to at most capital Q. So this is just a fraction here of an n tuple of rational numbers which are written with the same denominator. And you may ask now for points of this form which are close to M. M is something bounded so automatically then P will be also bounded by constant times Q and you may ask for the number of points which are close to M by distance at most delta over Q. So the delta measures how far you want to be away. Distance could be for example Euclidean distance. This Q is there for normalization purposes but in practice this doesn't hurt much. So that basically is like taking a shell around you manifold and asking for rational point which are close to there and that's closely related to the dimension close conjecture because if you take some algebraic variety and look at some alpha and patch and rational points on there then rational points on your manifold which say comes from the algebraic varieties exactly the same which you would count here for delta equal to zero. So you're kind of sticking the thing up and look at what's in here. That's also trivial bound. The first real upper bound you could get is of size Q to the M plus one. We may say about is a bit more later for example if R one, if you look in R one at an interval from zero to one then I would count here fractions P over Q one dimensional rational numbers. Fractions P over Q between zero and one and that's about Q square of them. This is what you see here. This would be a Q to the power one plus one. Sometimes there are that many points because you could for example choose as a manifold a piece of a rational linear subspace. Just the example we had takes some interval from zero to one in the line R one and then we will find actually Q square points on there. So this trivial upper bound of Q to the M plus one is sometimes realized for rational linear subspaces. This may be not a typical case. There's no alternative for realistic that I want to draw your attention to. So for a moment let's say M is the dimension of your manifold, R is the co-dimension of N is the dimension of the ambient space. Then you could try to understand this problem as counting lattice points. You could now imagine in this formula to first fix little Q that gives you capital Q choices. Now if I fix this little Q then I look at somehow a shell around M of well, sign up by delta over capital Q little Q is of size roughly capital Q for now for simplicity. This thing has co-dimension R so the volume that I get by sign up my co-dimension R manifold by delta over Q should roughly be delta over Q to the power co-dimension. So this delta over Q to the co-mine dimensions the volume in a shell I look at. And in this shell you then ask for rational points with the same denominator of Q. That's a lattice. So you look at if you fix that Q do you look at lattice and it says the determinant Q to the minus M. So if you naively think about this like taking the volume dividing by the determinant of your lattice you would expect in this shell delta over Q to the R times Q to the N points. And that's another factor of Q because we then in the end let vary Q. So if we calculate this we get the other to the co-dimension and Q to the dimension plus one. That is another characteristic that you may come up with. So maybe some point before we go ahead that rational linear subspace is containing many rational points is something special. If you take any manifold which is not a rational linear subspace similarly as in the dimension gross conjecture you may expect that there's far less rational points on there. So you might expect that if there's far less rational points on there then maybe if delta is somewhat large this contribution is the one that you could maybe see you in exhibit. But even then your manifold may still contain itself a reasonable number of rational points. There's just two examples maybe you can look at a second one for time reasons. If you take an N minus one sphere that's one equation say one squared plus XN squared equals to one. If you ask for rational points on there then by the homogenizing this is the same as asking solutions to the equation say Y1 squared up to YN squared is equal to say Z squared in integers. So again this is like the same question all I said we asked for a quadric value on and if I said your delta equals to zero and count the rational points on the sphere then you get back to constant again Q to the power N minus one points. So you expect on the sphere Q to the N minus one points and if you compare the data heuristic R would be equal to one in that case and M would be N minus one. So we would compare this delta times Q to the power N. Can you see that as long as delta times Q is larger than one this heuristic time should dominate the points which are on here. So we may expect that there's some lower bound for delta for which somehow in this shell around a manifold is large enough that say this heuristic expectation is the one that you actually count which is larger than the same the stuff that is going on the manifold itself. Nundi this is a conjecture which in this form has appeared in work of Yen-ni-huang I'm going to come to that in the next slide. If you take a bounded submanifold of our N boundary is proper curvature conditions this is actually regnet's supposed to be like that. So the question in the end under what conditions one can establish that conjecture then first of all one can conjecture that is there's a general upper bound which consists of this kind of heuristic how many things we generically expect plus a term which is the same that we've seen in the dimension cross conjecture. This is Q to the dimension plus epsilon is for the same reason here because if you think about an algebraic variety and I take an affine patch of this say projective variety and then take an affine patch and then could consider this as your manifold and then you may expect by the dimension cross conjecture maybe in the worst case Q to the n plus epsilon points on there. And if you then vary both the conjecture maybe that no matter what manifold which is somehow reasonably behaved has the same property that on the manifold you don't find more than these points and sort of yeah you could say like this should be maybe all the contributions you see the points on the manifold plus the thing that then adds up. You can of course then this compare those two terms so if you look at those two terms you see that as soon as delta to the r is larger than Q inverse or delta is larger than Q to the minus one over r then the first term is larger than the second term so you could conjecture that if delta has at least this size you may get this as an asymptotic delta to the r, Q to the m plus one if Q goes to infinity. For example if r is equal to one we get here again the bound delta is at least Q to the minus one which we've seen in the example on the previous slide. And there are examples like if a Markov that show that some curvature conditions are necessary even for every point like if you look at a Markov or D at least three then the point one zero zero one you have a vanishing Gaussian curvature and you get that many points which you could stick in delta close to Q to the minus one is gonna be larger than what would it be conjectured. This main conjecture has been proved in rather seminal work by one in 2019 for the case of hypersurface smooth compact hypersurfaces is the restriction that a Gaussian curvature is bounded away from theorem. Well, if we need some constraint here for example, yeah, rationally near subspaces we have we need to exclude for sure but maybe also other things like if a Markov and this is one on theorem that we now have in this generality. There was lots of previous work in this direction for curves Huxley was obtaining upper and lower bounds which were almost sharp then there was for C3 curves or C2 curves hence if you talk about upper or lower bounds works of Berezniewicz, Dickinson and Villani and one in the line upper bound first ones for the lower bound and also then a work of one for curves. So there's been a lot of building up in this direction but this was really a very milestone result and some very interesting consequences. That's why I started a talk in discussing the dimension curves conjecture that you can reproof the dimension curves conjecture for the varieties for which the corresponding hypersurfaces have a non-vanishing Gaussian curvature. This is because if I stick in in this conjecture delta equal to zero you get as an upper bound exactly due to the M plus epsilon if you rephrase this language from the algebraic varieties side to the manifold side this is exactly the same thing that you talk here about. This is very interesting because one's work is purely analytical. You throw away all the algebraic structure and you only remember that there's a manifold that you work with. And so in contrast to the earlier proofs using the determinant method this is very interesting that you somehow don't need any arithmetic structure at least for a certain class of varieties. Something that I want to touch upon for a moment is work in higher code dimension that a few years ago I was interested in together with Shintari Yamagishi which needs us to more general results. So a serious restriction is the cyber surfaces and we want to point out that one can do for certain classes more. If you say parametrize your manifold by r functions so say I take a vector x this is a vector say x1 to xm and then I take r functions this parameterizes a code dimension r manifold in r to the m plus r of dimension m maybe consider this thing locally in some ball then first step is you want to formulate this is a counting problem we are going to throw in smooth weights to make these things better behaved this may look like that. We look at this counting function here we count vectors a and set to the m and q up to capital Q so this a over q is the first m coordinates that you look in here for this x variable. And if I stick in for this x and rational point a over q I want to somehow that all my f i's evaluated at a over q are also close to rational point that you can formulate by asking for q times f i of a over q being close to an integer. This notation up above alpha for real numbers the minimum distance to the next integer. So if qf1 of a over q is less than delta for example it means that q times the first function f1 minus some integer b1 is an absolute value less than equal to delta on other words f1 minus b1 over q is less than equal to delta over q. That's exactly what we were looking for because this sort of says that the coordinate here is also close to a rational number with the same denominator little few. So is this really formulated and asking for vectors a over q which of these functions get well approximated. And here again we see it back another agenda heuristic but we expect how this may grow in a generic case. If I would drop the conditions of these functions f1 to fr and if w is some smooth weight function maybe think about just an indicator function of an interval for a moment then we would have the q to the n plus one points that we can't say little q runs up to size capital U and the others a run up to size q to as well. So get q to the n plus one and each of these integer parts lands between zero and a half so the probability that you hit delta is something like two delta and if these events are independent we should expect that say a proportion of delta that are often points actually satisfy these inequalities and that's the same heuristic that we had early on. In the case of a hypersurface say if r is equal to one and your parameterized by one function f1 then non-zero Gaussian curvature is in a given point x0 is the same thing as saying that a Hessian matrix does not vanish at that point. The class of many faults of one can extend this work of one which is more like a cloud. It's an interesting class with one example where one can do something is asking for the Hessian in the whole pencil of functions that parameterize your manifold never to vanish. It's a rather strong condition. One point out that he asked this in a real pencil in a complex or complex numbers it was not be possible so he asked in a real setting about this. So think about your Hessian matrix and take the linear combination of those then if you compute a determinant you end up for example with a power of a positive definite quadratic form and d1 to dr that would be one example that is would be realized. Such examples do exist for example from results of determinants of representations you can construct those but the nice thing is that one can prove again similar asymptotics and other bounds and the reason why I think this is a nice class on interesting class for one reason is that this conditions actually not as completely maybe random unless you may sing in first place also an analysis of one of my co-authors Wachler-Chevastava then later explained to me that also over there this is an assumption that sometimes you may like to use but a nice thing from an arithmetic point if you is nice that if you look at this class for which well there exist examples then we actually can count the number of points in a much thinner shell than we were conjecturing to. So the results that you get are actually stronger than what you would conjecture. So you conjecture that only up to q to the minus one over r you can count stuff but actually see we get a smaller at least if the co-dimensions, yeah, certain ranges. Well, even if there's something that is for example if the dimension m goes close very large and you actually approach delta as large and q to the minus one which would be the more the same for hypersurfaces which I think is kind of interesting because very often in a dimension gross conjecture you just project things down to the hypersurfaces there may be also sometimes certain conditions more going on in higher code dimension which would be suggesting by this example. So if we the reason why we can count in thinner shells is that we can show there's less points on the manifold that's why we can zoom in more because there's just less that could be disturbed and may encounter and display here the concrete upper bound it's maybe not, yeah, I can talk about what this is the trivial one, not a trivial so this would be the conjecture q to the m and we save some power. And again, if we think about the dimension m is being very large you actually say from r minus one here so for each code dimension you seem to save maybe you could save close to a power but I'm not gonna, yeah, in the detail here too much I just want to also maybe point out along the way of that it's not so easy to get away from the curvature condition that the corrosion curvature is not equal to a zero. So in higher code dimension even if this someone make this much stronger there's one very nice example by Rajschlaj Shrivastava Nick Klaas-Techno where they can remove for one point this corrosion curvature condition essentially spirit they look at hypersurfaces it's a kind of this shape which are given by a homogeneous function so they have function f such that f of lambda x is lambda due to some degree times f of x you could think about something like an L2 norm to a power of D as a concrete example and they manage yeah, so this by construction if D is larger than two it's gonna have a vanishing hasheon in the origin so you kind of construct examples that way that the hasheon vanishes exactly this one point in the origin that's the situation that they study that hasheon otherwise is non-zero and they need to obtain here upper and lower bounds of the sharp for the range delta if you inverse or the thing where you would want to study and something very interesting here is that there's another term that pops up in fact the result also holds for capital D being larger than n minus one and I would have to write it up slightly differently so this is one say sample result something interesting about their verse that they actually showed that if delta gets somewhat small and if the degree D if your homogenous polynomial is large then there is actually some other contribution that you're going to see from some kind of what you could call a cap set to think about a thing but just being very flat your hyper surface such that if you're zooming into the origin that somehow you just behave sort of almost that linear that has somehow many points in there that also shows that you can't just conjecture everything to go through without any further assumptions on curvature for example so I really like this work a lot because it brings a lot of interesting tools from the analytic side and sort of bootstrapping arguments which are very nice to have here but of course you may also want to go further and ask what happens now for even more general manifolds that these are all very nice examples that do something higher code dimension or for one point being maybe exceptional but what if you want to drop now basically all conditions in a bay maybe you don't want to look at a rational linear subspace we understand them but maybe one of the biggest classes that you could consider are what is called L non degenerate manifolds would say that if you have some manifold which again is parametrized here say by our functions and X now again an m dimensional vector so this m dimensional vector and our functions f one to f r you would say it's L non degenerate if the partial derivatives this vector of order up to L span or n in a point and say it's L non degenerate if this happens thing for all the points if you manifold completely this just means that your manifold is not contained in some linear subspace if you think for a moment about a hyper surface so one function f then being non degenerate means that there's just some derivative somewhere which is not going to be zero in every point that's really the biggest thing oh sorry derivative of all at least two that's not going to be equal to zero so in a way it's somewhat of a very very big notion that you could ask for and something surprising that together with Ratschler-Schwesterer and Niklas Teichler and we figured out is that we can actually do something about that class formally there's some counting function you would associate it we will stick in even more smooth rates even smoothen out the nominator and everything else but so sing about this counting function here as the same counting function that we have seen previously so this counts the rational points close to this manifold where the nominator is bounded by capital Q and you look in a shell of size delta divided by capital Q and under this very weak assumption of L non degeneracy we do manage to get an asymptotic with the constant the one that you would predict under that smoothing and delta to the RQ to the M plus one with some power saving in Q as soon as delta is not too small that we need a lower bound the delta is certainly expected this is most likely not a correct lower bound but it's some lower bound that's already very nice because I hope that I tried to illustrate a bit with the previous exam it's not so easy to just get rid of somehow the curvature conditions so in this context it's to our knowledge the first bound that gives an asymptotic in that generality yeah again sort of disbound while it gets bigger the larger dimension co-dimension or dimension and embedding dimension and also the non degeneracy parameters we can get lower bounds which are stronger for the lower bounds we actually get here the same constant so this constant I phrase it now here there exists a constant but what I really mean is we can produce a lower bound with the exact thing that you expect up to an error term which again is power saving in Q as soon as delta is at least as large as Q to the minus three over two and minus one there's been in 2012 rather groundbreaking work of Viktor Boris Nevich in publishing the animals in this case where he produced similar lower bounds for non-degenerate analytic manifolds so we could relax that a bit too smooth instead of analytic for delta larger than Q to the minus one over R so with that for the case that our depends on the dimension and co-dimension we actually get a stronger bound with three over two and minus one that's very nice because we can produce in some sense now even more lower bounds and also with different techniques we can do that there's also upper bounds you can stick all of these things together that we've seen before you get a uniform upper bound which incorporates the term that you expect and some well maybe not so nice looking term but what you get in the end if you balance these out and think about what can be the worst case being you obtain a bound for the number of points on the manifolds so that comes basically from saying that if I make delta smaller then also get less points so that there's a certain point for delta where at both times your balance and then this is an upper bound also for delta equal to zero because there's only getting more points if I think this up so there's a bound of the Jp plus one oh, okay minus one over two Ld times n plus one and here I made now here a yeah, I'm sorry about that I will correct that in the things to be uploaded this is the dimension plus one this is an n plus one minus something so we don't hear the dimension because conjecture is dead but we get some very general upper bound if you would think about projective varieties there's been some very nice work of a Schulte animoto where he gets for projective varieties some upper bounds though then you need to know something about the geometry even to compute the exponents and that's a nice aspect here that we get something of free now I'm going to say just a few words about a proof if you follow some or part of our proof isn't filed by a work of there is Nevischen Yang which I'm going to get to in a moment but the new part is using pro-assamation and introducing more free analysis into this thing we use pro-assamation we want to use things to studying oscillatory integrals the main strategy and using here already the language that there is Nevischen Yang introduce of generic and special parts in a way of previous work you try to some look at your manifold and there was some or there was there was Nevischen Yang introduced there were some special parts which are hard to handle which you have to somehow cut out and don't know what to do about this much very well and some generic part where you can somehow try to get up a balance we also do something similar but analytically we construct carefully a rate function to do this and then so for the generic parts we use even rapid decay estimates a bit more and we have to be a bit careful because things depend more parameters than maybe yeah your standard application maybe used to but with careful bookkeeping this works out and for the special part we use work of Bernick Kleinberg and Margules which already appeared in the work of Paris Nevischen Yang with quantitative non divergence which basically tells us that a set of the manifold where you have a point for which there exists some shift vector on a power saw and some which makes this integral somehow in a way that we can't control that this volume is small so basically tells you that there's not many points for which there exist shift vectors which are bad and this work of Bernick Kleinberg and Margules has already been also used in many contexts but for example in this work of Bernick Nevischen Yang so for us the new part is that we combine it with the free analytic side so for the last 10 minutes I want to discuss applications in diaphragm and approximation and see where also the work of that dimension of Paris Nevischen Yang which has been motivated from was not enough in the very beginning here again but directly lemma you can approximate a real number theta with a fraction a over q by at least one over q squared and multi-dimensionally if you have a vector theta one to theta n if you want to approximate it with a fraction a i over q simultaneously you can do that up to q to the minus one minus one over n you can also formulate that in terms of approximation functions if you're given a function psi and a point in Rn we say that it's psi approximable if there's an equality that a vector y minus a vector a over q in the infinity term norm is less than psi of q over q for infinitely many such choices that this infinity is supposed to be the infinity norm so we want that each coordinate y i minus a i over q is an absolute value less than psi of q over q for all i so for example by Dirichlet's theorem if you choose what psi of q function q to the minus one over n then you get here on the right-hand side minus one minus one over n that's exactly sort of what you come from this multi-dimensional Dirichlet lemma and Dirichlet's theorem would tell us that everything has infinitely many such approximations we can approximate every point that well you may now look at other approximation functions a very classical theorem in this area is Keynesian theorem which basically says if I look at some monotonic approximation function then there's a break of the pending whether the series of psi of q to the n is convergent or divergent so psi of q to the n is convergent which means some upsides rather quickly decaying so I want a rather very good approximation then u n is the the back measure of the psi approximate of the points in our n is zero so if you want to approximate your point so well that is psi of q to the n is convergent so then you make this very small here yeah then this is only for the back measure zero point at most possible on the other hand in the divergence case the complement has the back measure zero so almost all points are that well approximate if you would take Keynesian theorem and stick in again the approximation function q to the minus one over n then we are in a divergence case and so you would get that a complement has the back measure zero so almost all points are that well approximate while we even know better by Dirichlet's theorem but just want to show you that this is the normalization which we can which pops up here Keynesian theorem is basic application of Borel-Cantelli lemma which I just want to call quickly here so we are in a case that this approximation function has this convergence and if you then say look at a cube and of the n dimensional cube in our n and the psi approximately points in there you could ask the other question if I fix the denominator q which real numbers are close to a rational part with denominator q in this box zero one to the n so if I look at vectors a i over q and if you think this up by psi of q over q then you would look at such kind of it's a little boxes in each direction you're sitting up by psi of q over q and you take the union over all the a's this would be then everything that would be set of real numbers which are somehow close to one point with the denominator the little cube the measure of this thing is psi of q to the n because each little box has measured psi of q to the n over q to the n and you have q to the n such rational points denominator q but the sum of these measures of the eq is convergent and by Boyle Cantelli that means that for almost all points you can at most slightly and finitely many of those so the measure of those where you are contained in infinity many of these e sub q's must be 0 and that's exactly what we wanted to show we wanted to say that for most points there can't be infinitely many approximations if you're in a convergence case so I'm showing this argument to you because I want to point out that what we use here is basically we are able to count rational points of denominator equal to q in a box and if you now want to do approximation not in rn but close to a manifold that's where then the results of counting rational points close to a manifold come in very much for that reason a question has been phrased in different variations different works that I named here it's been a following that if I take a sub manifold in rn and again a monotonic approximation function for which I assume this series to convert to diverge then you could ask if also restricted to the manifold almost all or no points are psi-approximable so imagine you take an approximation function for which this thing converges then you know that in rn the lag measure is zero of the points that you can very well approximate but it could be that maybe they are all somehow contained or that in your manifold it could be that your manifold behaves exceptionally that despite an rn you can approximate that well typically you can on m still also the other way around and so this question basically asks that if I restrict to a manifold does this still behave somewhat the same way does a typical point on rn behave like a typical point on a man-vehicle the divergence case has been essentially proved for smooth manifolds from the general by the work of barris-nevich that I mentioned earlier on on lower bounds and rational bounds and close to manifolds and very recently the convergence case has been proved by barris-nevich in young exactly using other bounds for rational points close to manifolds and the reason why you need here other bounds for rational points close to manifolds is exactly the argument here by Borel-Cantemini more or less if in total they are not for each denominator Q or h of denominators 2 if they are in total not too many rational points then there can't be also very large measure of real points which are close to these rational points so this is a direct application and with our work we can actually re-produce that theorem so our upper bounds that we obtain for rational points close to manifolds are stronger what is produced in there but maybe because we've managed to kind of not in the free analytic techniques on that side that is not visible in re-proving that theorem but it's visible if you ask finer questions in this direction if you say take some subsend to find dimension as a host of dimension and write this what host of measure HS then you could ask if I look at some approximation function Q to the minus tau and for example for tau is equal to minus 1 over n this is what's covered by Dirichlet's theorem host of dimension should be n everything Rn is that well approximate but if I make tau a dull large so if I want to approximate better then I know that a Lebesgue measure is going to be equal to 0 of the things that I can approximate well but maybe the host of measure is somehow still non-zero you could ask about a host of dimension of the things that are somehow even better approximable and in Rn one can solve this work of Janik Bezikowicz for large tau's for just Rn which again is sort of a very similar concept as to what you've seen in the Kimchins type theorem and you can now ask a similar question for a manifold so if you take a manifold and look at the points which are very well approximable as n of psi you somehow want to approximate very well can you still find the dimension the host of dimension and there's a question or conjecture I will move to this slide this appears in Barysimic in Yang as well so they say that maybe by a similar heuristic of volume computations so this is a very natural thing here one can compute a dimension the host of dimension of these very well approximable points for certain values of tau and there's a certain limit where this may not work anyone has also good reasons for this and one question is now how large can we make this tau so to what extent can we still understand the host of dimension so the larger tau gets the better we want to approximate the points and so it's a challenge to ask like how much large can I make the tau how well can I approximate and still understand at least the dimension of the things that are that well approximate and given that we get stronger upper bounds by our techniques we can improve on that aspect by well that's something neither do the work of Barysimic in Yang give the conjecture now of course do we but a range of say these values of tau such that we can find a dimension that gets larger there's different inequalities and we're not so illuminating but I just want to say that sort of information that we have on rational points close to many false gifts sort of here of information on about a fan and approximation so I think now we are maybe just in time so thank you very much