Emmanuel Kowalski: Wasserstein metrics and equidistribution (NTWS 292)

This is joint work with Teo Antro. And as the title says, so it's about equidistribution. And especially quantitative aspects. So, let me first remember recall the definition of equidistribution in the setting I'm going to work on. So, very abstractly equidistribution is some version of convergence of measures of probability measures on space. And the version I'm going to look at will be when you have a locally compact topological space, very often compact, but it could be something like the quotient of the upper half plane by a discrete group which might have finite volume and not be compact. We have some probability measure mu on X which is kind of a reference measure to which other measures would like to converge. So, mu n will be a sequence of measures of probability measures again. Everything being happening on X. And the example to to keep in mind for for most of what I'm going to talk about, the standard example is when you have a finite subset non-empty in capital X and then you look at the measure which is just the average of delta masses over this finite subset. That's the standard example and then we say that mu n becomes or is mu equidistributed as n goes to infinity. Uh by definition means that you can compute the integral of suitable functions with respect to mu on the on the set capital X as limits of the corresponding integrals with respect to the measures mu n. Okay, so and in the example this means the limit of the average of the test function f at the points in the finite subset that are varying. And of course this statement for an arbitrary function f is way way too strong to ask in general and so one imposes restrictions on the test functions and classical definition for equidistribution is to assume that this works for f continuous and bounded. So if if x is compact this is just f continuous if f is not necessarily compact then we also uh assume that it's bounded. Okay? So this is a classical definition uh let me give one or two examples uh for those who might not uh be very familiar with it. Uh one of the most basic ones uh goes back to Hermann Weyl uh in the early 20th century is uh taking the space X to be the circle or R mod Z. Uh the measure mu to be the Lebesgue measure. And for instance the xn so Hermann Weyl proved many statements of equidistribution of this kind but the most basic one would be when you take uh the measures mu n to be defined as before by averaging over a finite set and the set xn would be the set of uh multiples up to n alpha for some element alpha which is in uh R minus Q. Okay, and of course alpha, 2 alpha, and so on have to be taken modulo Z. And then it's a fact that mu n or xn one could say is indeed uh equidistributed according to Lebesgue measure when n goes to infinity. And it's a prototype of many statements of equidistribution. But just to give um so this is about I think 1917. And just to give another in fact uh even earlier example even if it's was not interpreted in this way at the time. So if you think of uh Dirichlet's theorem on primes in arithmetic progressions, it can be interpreted as equidistribution. In this case uh the set X is just finite. It's the invertible classes modulo some fixed modulus Q. Uh the set xn again so the measure mu is the uniform probability measure on this finite set. And uh xn could be the set of uh primes. So you look at the uh set of residue classes uh P modulo Q. Let's say for primes P uh up to up to n. And you need P to be coprime with Q if you want to literally for this to be inside capital X. Okay, and then Dirichlet's theorem says that then there is equidistribution. Meaning that every congruence class modulo Q can be represented equally often, roughly speaking, by primes reduced modulo Q. Okay. And this example is a good example to see that it's very important very quickly to not only be able to say that equidistribution holds in the sense that these various limits that I've stated here exist and are equal to what they should be, but you also want to have more quantitative information. And that's for anybody who has studied any multiplicative number theory one knows very well that Dirichlet's theorem on primes in arithmetic progression is just the beginning and one really one needs to have error terms and quantitative information. Need to have quantitative statements in many applications. So I'll give concrete example of such applications in quantitative terms, but for the moment what do we mean by quantitative? So I'm going to stick to to settings which are closer to the results like Armand Weil's than to Dirichlet's theorem. And so one of the easiest way is to say well we we restrict the set of test functions and we try to say that the left hand side of the equality, the integral with respect to the limit measure, is equal to the nth term on the right hand side plus some explicit error term. And if you do this, there are many cases where you can do that, but it is not necessarily what's needed in applications and for instance maybe what most people most classically associate to the notion of quantitative equidistribution is the Erdos-Turan inequality. So this is the maybe the first example which has been studied for a long time. So the the first version of this by Erdos and Turan is from 1948 to give an idea of the time scale. So this concerned the case which is the one considered by Weyl. So the space is the circle the limit measure is the Lebesgue measure but it can apply in principle to any sequence of measures which can approximate approximate mu and it quantifies equidistribution by comparing the measure of intervals for the Lebesgue measure so the length of an interval with the corresponding measure for mu n. So in the case of the example so this would be counting how many elements in your set belong to the interval. So how many elements in xn belong to a given interval. So equidistribution in the case of the Lebesgue measure means that this goes to zero when n goes to infinity and the way it's quantified by Erdos and Turan is to take the supremum of all the intervals. And the interval you can think of closed intervals or open intervals it doesn't matter. And they find a quantitative upper bound for this uh, depends on what are called the Weyl sums for equidistribution. So, the upper bound is of the following form. So, it's bounded up to a multiple constant which can be made explicit by 1 over T. So, T is going to be some parameter which can be a real number that can be chosen arbitrarily and usually is chosen to give the optimal possible bound. Okay, so maybe I should make clear that when I write this less less symbol, it's the uh, it's the Vinogradov symbol, so it's means that it's bounded by a certain constant times whatever follows. So, in this case, so 1 over T and then a sum over integers H in Z with absolute value non-zero so, H is non-zero but bounded by this capital T. And then I have something which decays with H, 1 over the modulus of H times the modulus or the absolute value of this Weyl sums which I'll call WH of μn in general uh, which are in terms of integral, this would be the integral of the function exponential 2iπnx uh, 2iHx with respect to μn. And again, in the case of the example that is just averaging the additive character exponential 2iπHx over the finite set Xn. Okay. So, if one looks at this inequality, uh, then it reveals quickly that, uh, to have equidistribution uh, is equivalent to having this WH of μn go to zero. This is an old fact that was established by Armand Weil uh in the same paper that I mentioned earlier. And uh this shows that if you know good bounds for these so-called Weil sums, then you will be able to provide an estimate for the supremum of this uh difference between the Lebesgue measure of an interval and how many elements in the finite set, for instance, happen to be in that interval. So, this is something that has been used uh many many many many times, and it's uh quite convenient in its way. And uh what I want to discuss is uh problems where this is not actually a such a satisfactory way of uh quantifying equidistribution. So, uh issues with this Erdos-Turan inequality. Okay, so for certain purposes, it's perfectly fine, uh but there are others where uh it might not be what what one wants. Um So, I'm going to highlight two of them. So, one is uh higher-dimensional versions, uh which feel less intrinsic, at least to me. So, I'm not going to state any uh of these higher-dimensional versions. There are many classical ones. So, the thing is usually there what happens is you replace the intervals by a certain choice of types of subsets, and uh if you look at a higher-dimensional version, meaning in R mod Z to some power, uh it's not necessarily clear why, for instance, rectangles should be better than other subsets, depending on what you want to to use this for. Um And in in our case, in this in this work I had with uh Thai Van Vu, the issue started there because we had equidistribution not in R mod Z to the power D of certain limit measures but we had the measure on a on a sub torus and we could not there was no canonical coordinates on that uh on that subgroup and so it was not necessarily very natural to choose any choice of coordinates to define the rectangles that you would use for a classical uh I don't remember the Erdos-Turan inequality. So that's one potential issue. And the other one which is somehow related is what I will call lack of invariance under various transformations. And what I mean here is that suppose uh we're in the situation of the Erdos-Kac theorem and you have some kind of other function G uh from R mod Z to R mod Z uh which is continuous. Then uh if mu n converges to the Lebesgue measure mu then it's a completely formal fact just looking at the definition that if you push forward uh mu n by the function G it converges to the push forward of the measure by the function G. Uh so that's completely formal at the level of equidistribution but the problem is you also might want to have such a phenomenon or principle uh of transformation that also applies for quantitative statements and if you measure the convergence of mu n to mu using the discrepancy in the Erdos-Turan inequality and you apply uh some continuous function well there will be first the problem that the limit measure G lower star of mu is not equal to mu in general. And And the Turin inequality is specifically written for the Lebesgue measure. And even if you took a function G so that this is again the Lebesgue measure, it's not straightforward just looking at the the statement of the Turin inequality what's going to happen for the discrepancy of the image under the function G of the mu ends. Okay. So, this is what we we were looking in in in our application with Theo, what we were looking for was a more abstract or maybe a more invariant way of measuring quantitative equidistribution. And it turns out there's a way of doing this which is uh in some sense was staring us very obviously so uh in the face because it's an extremely well-known uh way of measuring the distance between probability measure. It's this uh Wasserstein metrics. Okay. So, they're also known as Kantorovich metrics or um uh Kantorovich-Rubinstein. There's There's various names, but the one that seems to stick the most is is Wasserstein metrics. Uh so, it's a very important concept in uh analysis. Uh especially in optimal transport. But it's also very often used in modern probability, statistics, uh actually many many many many fields. Um And as I said, it's it's a way of giving a distance on the space of probability measures uh on on suitable spaces. So, I'm going to give the definition. As I said, this if you've ever seen a colloquium in analysis in the last 10 to 15 years, there's a good chance that optimal transport was mentioned, and therefore there's a good chance you actually saw this definition. Uh because it's really become one of the uh most uh exploited tools uh in many many parts of analysis. So, I'm going to again work in less generality than and then these metrics are defined. Uh so, one small difference with the setting I was before is now we have to we need to have a metric space. For applications, this is usually not an issue, but it will depend on this distance D. Uh the metric space has to be complete. But it need not be locally compact. There's There's many cases where infinite-dimensional spaces are used in this setting. Um so, we're going to define distances between uh probability measures I'll call them just mu1 and mu2. Probability measures on X. Uh these will depend on a parameter P. So, it's a real number. Okay, so at some point later P will become a prime number as it should, but for the moment this is a terminology that's also so standard that using anything else would be a bit problematic. Um so, we have a parameter which is this real number, and one defines the Wasserstein P metric between mu1 and mu2 in the following way. So, it's an infimum over measures on the product in a set that's called capital pi of mu1 mu2 usually, which is defined as the set of probability measures. So, I'll call them pi on the product of X with itself, such that if you project with the first on the first coordinate you obtain mu one. And if you project on the second coordinate you obtain mu two. Okay? So, for instance, the product measure of mu one with mu two will have this property. So, this set is always non-empty. And once you have this, then you integrate over the product the distance between two points to the power P with respect to this measure pi and you take the power one over P. And then you take the infimum over this. And in this generality, this infimum is actually achieved. I'm not going to use that fact, but it's something that's worth knowing. Okay. So, it's a definition that one finds now in books dealing with optimal transport, for instance, and so on. And it turns out to have many, many good properties. So, the first, the basic one I want to mention. So, all together, there'll be a list of, I think, four properties and they they really show that this gives a good quantification of the notion of equidistribution, of convergence of measures. So, the first one is that it does metrise a convergence in law or equidistribution, at least if X is compact. So, in the more general case, you would need to have some finiteness condition on the on the measure, but in that case to say that mu n converges to mu in the sense of equidistribution is equivalent to saying that the distance between mu n and mu goes to zero as n goes to infinity. So, any quantitative bound that goes to zero for the Wasserstein distance would give some kind of quantification of equidistribution. And this holds for every fixed P. So this does not depend on the choice of P. Uh that's the first property. The second one is it has these strong invariance properties, which is what I was saying the Erdős-Kac inequality does not really have. Uh so it doesn't quite work with an arbitrary um function that's continuous uh between a space and another, but it works for Lipschitz functions. So suppose you have a Lipschitz function from X to not necessarily the same space, let's call it Y. And let's say it's Lipschitz with constant C, so it does not distort the metric by more than a factor C. Okay, so Y is another metric space. Then uh it is completely formal that uh again for any P uh computing the distance after taking the push forward by uh the function G uh cannot increase the Wasserstein distance by more than the factor C. Okay. So that means any kind of quantitative statement you you might have to measure equidistribution in Wasserstein metric anytime you apply any Lipschitz function, this will translate into uh essentially uh equivalence equi- equidistribution quantitative equidistribution on the target space Y. So what you're losing is this Lipschitz constant. The inequality being with no implied or mysterious things means that it's also actually useful maybe when C depends on other parameters, and it will give you something uniform uh without having to do anything. Okay? And this invariance property is, as I said, completely straightforward from the definition. There's essentially nothing to prove. Which for me is a good sign that this can be useful for for the kind of things we want to do. Um Then the third uh is uh specific to the case P equals 1. There's there's a version which applies to other values of P but is uh less straightforward. So when P equals 1, this is something called the Kantorovich or Kantorovich actually, I think. uh Rubinstein duality. And it's a very nice uh functional interpretation of W1. So W1 mu1 mu2, the distance between the measure mu1 and the measure mu2 if you use P equals 1, it's the same. So the as a supremum, so it's a kind of a duality between min and max. But now you take the supremum over uh continuous functions which are one-Lipschitz. And then you compare the integral of F from mu1 and mu2. Take the modulus and take the supremum over all one-Lipschitz function. So it's a kind of a functional uh interpretation of the W1 metric uh which is obviously extremely useful and and extremely nice and pretty. So when P is larger, there's a similar statement but the set of test functions becomes uh much more intricate. Um One might say that this suggests to actually define W1 by the right-hand side instead of this uh play with measures uh projecting to mu1 and one and mu two uh and it's true that for what I'm going to say today you could do it this way. You could take the right-hand side as the definition. Uh and forget about all other Wasserstein metrics, but in the general theory it seems really that actually P equals one like an L1 space is is pathological in some respects. And it's actually better to uh to use uh other values of P like P equals two in particular. Okay. And the last one uh the last statement is again uh even more restricted. So, I'm going to take P equals one. Uh and the space X will be very special. This will be a compact uh connected Lie group. Okay. An example is this R mod Z to the power of D that I discussed before, but you should also think of uh unitary matrices or special unitary matrices or special or unitary symplectic matrices of some size. Uh and in that case when you take D to be a suitable uh uh invariant metric. So, in the case of the torus it's the usual one in the case of SUD or SUSP2G, these are more of strike metrics defined in the theory of compact Lie group. So, I should say pi invariant metric. Um then uh there is a theorem uh due to Bobda in the general case and uh Bobkov and Ledoux for tori. So, for the first case. Uh Uh and this inequality will look like an Erdos-Turan inequality, but for W1 on such a space but very much similar to the Erdos-Turan inequality in some sense. So, it says that uh for mu1 and mu2 arbitrary measures, probability measures on on this group so I'll call it capital G cuz a group should not be called capital X, I think. So, let me get my statement to be sure I get the right one. So we estimate W1 of mu1 and mu2 it's going to be bounded by a constant depending on the group by something like again 1 over T where T is a parameter that can be optimized. And then instead of a a sum of Weyl sums, it's now going to be an L2 kind of average. So, it's a sum over parameters uh lambda with norm up to T. I'm going to say what these are in a second. Uh then there's the dimension of the associated representation divided by another quantity that I'll define which is kappa lambda and then uh the Fourier coefficient for associated to lambda for mu1 minus the one for mu2 squared. Uh this is a Hilbert-Schmidt norm and everything is raised to the power 1/2 because this is some kind of an L2 uh right-hand side. Okay, so I should say what all these various things are. Uh so, the lambdas are the highest weights parametrizing uh irreducible representations of G. So, in the uh case uh of R mod Z to the D, which is already very interesting, then uh lambda is just an integer in Z to the power D. Um So, in general, these are elements in the cone in a cone, sorry, uh contained again in a lattice of some dimension uh related to the to the Lie group. Uh then we have So, the irreducible representation associated to lambda is denoted by rho lambda, and then it's the dimension of the underlying space. So, in the case uh of lambda in Z to the D, then rho lambda is a one-dimensional representation, uh and it's defined by rho lambda of X equals exponential 2i pi the classical inner product of lambda with X. Um And the dimension is just one. Then the kappa lambda is so-called uh Casimir eigenvalue or Laplace eigenvalue. It's essentially the variant of the Laplace uh uh an eigenvalue of a variant of the Laplace operator. So, in the case of R mod Z to the D, the norm of lambda is just the Euclidean norm, maybe up to a constant multiplicative factor. Uh and the final thing is this Fourier coefficient. I will define them here. So, the mu tilde for an arbitrary probability measure mu on G, and for an arbitrary lambda, uh, what you do is you integrate uh, over the group with respect to the probability uh, measure, uh, sorry, no, with respect to the measure mu, the image of uh, rho lambda of an element, you take the adjoint, and you integrate, as I said, with respect to mu. Okay? So, this is a a linear map on the space of rho lambda. So, it's a linear transformation, which is why here we have a Hilbert-Schmidt norm uh, to compute the the size of these things. And in the case of the torus, well, this is just the usual Fourier coefficient. It's a one-dimensional thing. It's just the integral over R mod Z to the D of exponential 2i pi inner product of lambda with x with respect to x. Okay. So, that's the uh, this inequality of Bourgain, and I should say that recently there's been uh, a preprint by Bourgain uh, Gronin, which uh, states a version of this for other values than p equals 1. So, we haven't yet incorporated that into our application, but this should be straightforward, and this should be uh, very useful to obtain applications where uh, you use WP where P is not equal to 1 uh, necessarily. But for us, we started with p equals 1, and this inequality, as I said, is highly comparable with the Erdos-Turan inequality, of course. Uh, in the case of the torus, the uh, quantities which appear are exactly the same as in generalization of the Erdos-Turan inequality, uh, so the bounds you obtain are kind of very similar to what one would get in any application of the other student inequality, but what you bound is not the discrepancy, what you bound is this Wasserstein metric. And then you can exploit the invariance properties of the Wasserstein metric to to go towards applications. So, let me now in the remaining time discuss some application. So, I'm going to present the one that we have in our paper with still, but there are others that have already actually been done by a few people since our paper was on archive, and I'll mention Cornelissen. So, Sief also cuz this kind of show how this really can give a very uh flexible uh framework for equidistribution, or can Bring a ling. And there's another by Peter Humphries, which does the kind of Duke's theorem or equidistribution on the hyperbolic on quotients on the hyperbolic plane in terms of Wasserstein metrics. Okay. So, the the application we have is a version of Deligne's equidistribution theorem, which is effective and quantitative. Uh and here again, I should say this is not the first time that people uh quantify Deligne's theorem. It's very natural to try to do this because Deligne's equidistribution theorem is proved by very strong bounds coming from the Riemann hypothesis on finite fields for the underlying Weyl sums, and therefore it in some sense was already from the beginning a quantitative equidistribution statement. So, here I want to refer to an old paper of Fourier and Michel, which is very much in the spirit of what I'm going to describe, so from 2002, which kind of did by hand some of the things I'm going to describe, and more recent work by Fu, Lau, and Chi, probably 2020, 2023 or 2024, who have a fairly general version in terms of some version of Deligne's equidistribution theorem. So, Deligne's equidistribution theorem is a very very general statement about existence of limiting distribution for families of exponential sums over finite fields, and since I don't want to assume the kind of background material that's involved in a general statement, I'm going to do a special case, which is already quite interesting and important for applications. So, this is going to be about hyper-Kloosterman sums. So, we fix an integer R. Now, we're going to from now on P will be a prime again, and the the Vassiliev matrix will all be in W1, then for a prime number P and an invertible element modulo P, one can define, following Kloosterman and and Deligne, the so-called hyper-Kloosterman sum, KlR of A and P, which is some normalizing factor, 1 over P to the power R minus 1 over 2, and then you sum over R elements in FP, restricted by the condition that the product is equal to A, and you sum the additive character exponential 2 i pi over P applied to the sum of these elements. Okay? So, when R is equal to two then you have one over square root of P and then you have X and Y where the product is equal to A and if you express, let's say that X2 is A over X1, then you recognize this is a classical Kloosterman sum. In general, this is a generalization of Kloosterman sum which occurs naturally in many problems of analytic number theory. And uh the distribution properties of this was established uh so, by Deligne and Katz. Uh so, let me state it in two steps. Uh so, Deligne proved um around 1974 that uh for every A and for every P, you can find uh matrix that I'll call theta of A and P uh unitary matrix uh of size R minus one uh no, sorry, of size R of C. Uh actually, with determinant one, but let's say just a unitary matrix of size R such that uh such that the trace of this matrix is the hyper-Kloosterman sum. Okay? So, this is already an extremely strong fact because it immediately implies, for instance, because the trace of a unitary matrix is bounded by the size that the hyper-Kloosterman sums are bounded by R. So, So you take R equal two, you get you get the Weil bound uh for the classical Kloosterman sum. So this is a special case of the Riemann hypothesis over finite fields. Uh already quite strong. So RH over finite fields. Okay? And uh it follows from the work of Deligne and then from the the work of of Katz. So Deligne more or less proved there has to be some kind of equidistribution property for these matrices, and Katz actually determined precisely what the equidistribution properties are, and they can be phrased as follows. So if you take uh uh Okay. So there exists uh a co- uh uh a Lie group, compact Lie group G sub R such that the uh if you take for a given prime P all the various matrices associated to these uh to this prime uh and they depend on R, I should say. Cuz the rank is important. Uh so there exists a Lie group, which also depends on the on the R, such that these uh become equidistributed. Uh And in fact, not in the group itself because I didn't say it, but this is the space of conjugacy classes. Cuz these matrices are only really well defined up to conjugation. Uh but uh these matrices become equidistributed in the space of conjugacy classes uh with limit, so the the measure mu in the equidistribution uh equal to the uh image of the probability our measure. So, they are uniformly distributed. That's the way to think about this. And moreover, so in some sense, this statement was already known to Deligne uh except for some subtleties. But what Katz did, which is the crucial to actually understand what the statement means, is to uh compute the group, and he showed that the group is the space of unitary matrices with determinant one when R is odd, and it's the space of uh unitary symplectic matrices of size R when R is even. So, this is what the Katz called uh or calls the the monodromy group for this family of exponential sums. Okay? And so, the uh the first application uh that or one example of the general form of the equidistribution theorem that we proved uh with Theo for uh Wasserstein distances uh is the following. Um so, that uh the W1 metric between uh I should give a name for this probability our measure. Let me call it mu sub uh R small R. Uh so, the distance to the limiting measure associated to the hyper-Kloosterman sums with the integer R, and the sampled, so I'll write it explicitly, the average of Dirac masses associated to these conjugacy classes, viewed as elements, everything taking place in the space of conjugacy classes. This is big O of P to the minus one over the dimension of this monodromy group. And this is the statement that we prove in in full generality for any family of exponential sums where the monodromy group is also connected. It's also it's always going to be a compact group. It's not always connected. We're working on on generalizing the statement to non-connected groups which definitely should be possible. Okay. So in the few minutes before end I want to emphasize two two things about this this result. So one is a concrete application. So what does this tell us about hypergeometric sums that we might not have known before? So one typical application of quantitative equidistribution is what people call often shrinking target problems. Which means you're trying to compute how many let's say of these conjugacy classes are in a certain set where the set is not fixed independent of the prime in this case, but actually becomes smaller and smaller when the prime grows. And here so what this gives us is the following theorem. So I'm going to state it when R is odd because the the statement takes slightly different form depending on the on the parity. I'm going to take then a matrix G0 in SUr of C. So that's the group GR for R odd uh, at least three different eigenvalues. Okay, so in particular, uh, it cannot be the identity. Uh, and then uh, we are able using the Wasserstein metric to get uh, lower bound for the number of uh, hyper Kloosterman sums which are close to the trace of G0 uh, which are close by a constant so let's say one divided by P to some exponent which turns out to be three times R squared minus one. Exponent is not so important. Okay, so you see if we just were saying that we want this to be less than some fixed epsilon we would just want the Kloosterman sum to be close to the trace of some element in SU2 of C and equidistribution will tell us that this happens uh, at least once and with the right proportion when P goes to infinity. Now we have uh, a distance between the hyper Kloosterman sum and the target which is shrinking when P goes to infinity and we show that this is bounded from below by for P large enough by uh, a constant times P to the power one minus two over three times R squared minus one. Okay, so we show existence but we also show uh, that there are in fact uh, many of them and what is relevant here is that this is the right proportion. In the sense that this is the our measure of the set of matrices where the trace of G minus the trace of G0 is bounded by this approximately up to constant. Okay? So, that's something that it's a statement which does not mention uh Wasserstein metric or anything. Uh and the way it's proved is well, we start with the Wasserstein bound uh that we that I stated before. Uh this is only provable in a kind of easy way because of the Riemann hypothesis and Bott as inequality. But then the trace map from the conjugacy classes to C is a Lipschitz map. And therefore, uh we obtained a comparable quantitative bound for the distance between the image of these measures by the trace map. And then we have to do some uh construction of test function to deduce uh this this type of things. And I'll take one more minute uh with one last remark which I I find interesting. Uh I'll I'll say it relatively imprecisely, but uh one can show that assuming a bound uh like W1 between the R measure and this average uh let's say suppose you assume that you know a bound P to the minus alpha. So, I just showed that it can be proved, but suppose we didn't know the Riemann hypothesis, then we could not prove it. But suppose then that you say, "Okay, someone gives you a bound like this for some alpha strictly positive." Then from this you deduce uh zero-free strip. Not not the Riemann hypothesis, but a zero-free strip for certain L functions over the finite field. Associated to the uh to the hyper cluster sums. Okay? So, I'll stop here, but for me this is an interesting point because it shows that somehow uh bounds in Wasserstein metric, they do contain uh some of the most relevant uh arithmetic information uh that that go even into the proof of equidistribution. In this case, it's not the Riemann hypothesis, but the zero-free strip is already something very very strong uh and I find that intriguing. Yeah. So, I'll stop here.