 Thank you very much, Alina. Thank you for the introduction and thanks for inviting me to this wonderful webinar. And I apologize for the typo in the year. We are still in 22. Thanks Mike for spotting. Hopefully it's the last typo in the top. So today I would like to talk about a recent new phenomenon discovered by Peter Sarna called optimal strong approximation and a way to prove it using the Sarna pre-density hypothesis. The optimal strong approximation result is a new type of result which sits in a natural progression of results starting from strong approximation passing through super strong approximation which I would like to tell you about. These are results about arithmetic groups and I thought I'll start with perhaps the prototypical example of an arithmetic group SL2z. So SL2z, I'm thinking about it as a group scheme over the integers that means that for any ring commutative with unity I can assign the group SL2r, the group of two by two matrices with coefficients in r whose determinant is one. And for example, if r is the ring of integers, then I get the arithmetic group SL2z. If r is the ring of integers modulo q for some integer q, we get the finite group SL2z mod q and there is a natural map going from the arithmetic group SL2z to SL2z mod q, which is the taking modulo q in each coefficient. And the natural question is whether this map is onto and it is a classical result and it's a special case of the phenomenon called strong approximation which says that this is in fact onto. Honestly, I don't know who to attribute these results. I'll be happy to hear if there's an expert in the audience who can tell me who was the right person, the first person to actually discover it. But the proof is very elementary, very nice. It uses the Chinese remainder theorem to reduce to the case where q is the prime power and then you just use Gauss elimination to show that any matrix in SL2z mod q, you can write it as a product of elementary matrices and those one you can leave to SL2z. Let me try to reinterpret this strong approximation result in a more geometric graph theoretical form and to do that I need to recall the notion of the construction of Cayley graphs. So let's fix once and for all a finite symmetric generating set of the arithmetic group and then for any integers I can I denote the finite group gq to be SL2z mod q the finite subset of it sq to be the image of the generating set of the arithmetic model q and to this I assign the Cayley graph that's the graph whose vertices are the elements of the finite group and I adjoin to each vertex g vertices of the form g times s for any s in the finite set sq. And it's easy to observe that the the the graph xq is connected precisely when the modulo q map is subjective and therefore we have an equivalence reformulation of the sum approximation result as the fact that all of these graphs are connected. So this is the first property and next I want to tell you about a notion coming from theoretical computer science and graph theory which is a strengthening of the notion of being connected. So a graph these are connected or not but the computer scientists are also interested in graphs which are highly connected and these are called expander graphs and they are measured using this serial constant so what is an expander graph essentially it's a graph that in any way that you try to separate it into two steps a and the complement of a you have to remove a lot of edges in doing so you know if you want to separate it and an example of a non expander is of course you take the two complete large graphs and you connect these two clicks by an edge this is a connected graph but it is poorly connected. Now with this new notion of highly connected graphs these expander graphs one can state the stronger property which is in the literature it's called super form approximation it's the property that the key graphs are not only connected but they form a family of expander graphs. So this is in fact a theorem and it was you can prove it in two ways the first proof uses actually the famous Selberg 316 theorem in which he showed the spectral gap for the Laplacian acting on Riemannian surfaces which are the upper half plane divided by certain congruent sub-groups and it's not surprising that the k-legraphs are in some sense closely related to these Riemannian surfaces and so Selberg 316 theorem could be used to prove a spectral gap for these k-legraphs and these this spectrograph is in fact equivalent to the notion of expansion. So this is the first proof of this result there is now a second proof which is more elementary and it goes under the name of the Bougain-Gambert expansion machinery and I'll get back to this result in a second. So we have this fact that the k-legraphs are expander and I don't want to to to survey the notion of expander. To those of you who never heard about expander let me just say that they've been intensively studied and they are they have remarkable applications in computer science and also in pure mathematics and the reason why they have these many applications is because expander graphs display several extremal properties through the randomness excellent mixing behavior and if you want to learn more about them there are a world two award-winning surveys in the bulletin of the AMS about expander graphs and I will just want to to focus on a single property that expander graphs display and that property will also be important for a lifting problem natural lifting problem one can ask for these arithmetic groups. So expander graphs have the the extreme property of having logarithmic diameter so the diameter of the graph is just the maximal distance between any two vertices of the graph and any regular infinite family of graphs their diameter cannot go below log the size of the graph and the expander are actually you can show that their their diameter is bounded by some constant depending on the expansion times the log the size of the graph so they're they're excellent they have the logarithmic diameter in this sense. Let me also note that the diameter you can actually interpret it as a property or as a bound on a natural lifting problem so first things first because the graphs are symmetric the diameter the the maximal distance between any two vertices of the graph is the same as the maximal distance between any vertex and the origin in the graph right and now you can reinterpret the the diameter as saying that you take the maximum over all the elements in the finite group and then you minimize the word length of any element in the lift of that finite element g so this is a this is a bound on the lifting problem in terms of the word metric and this is brings us right to the setting of Sarnac's results which is an attempt to state an optimal version of the lifting problem and to do that I would I would conform to Sarnac's notation and and instead of measuring the size of elements in the arithmetic group by the word metric which is a bit arbitrary in unnatural because it depends on the generating set instead I will I want to measure the size of element in SL2z by the norm and in is so using this norm I want to introduce the following almost covering exponent constant which Sarnac described in this letter I'm going it's a nested definition let me start from the individual covering exponent of a specific element g in a specific parameter hue so there is this two over three constant appearing so start from right to left so I cannot and you'll see why there is this normalization of two over three but other than that the individual covering exponent of g is again you take the minimum over all possible lift of g and you what you're minimizing is over log to the base of q of the norm and and then you can take the mean covering exponent of the finite group of the parameter just averaging over all the individual covering exponent and finally you take the lim soup of this covering mean covering and this is called the covering exponent colloquially you can Sarnac describe it is you can think of the set of elements in SL2z of a norm bounded by some parameter t as a set of your balls and the the set of element that the finite group SL2z mod q is the set of boxes the model q map is the map sending balls to boxes and the this the kappa mu is the optimal constant exponent such that whenever q square is slightly over q to the three times this exponent then almost every box contains a ball okay and okay so now I can also explain why is the why is there this ratio two over three the reason is the number of balls grows like two square and the number of boxes grows like q to the third that's why there's the three over two over three normalization and clearly by a simple pigeonhole principle the the almost covering exponent cannot cannot be smaller than one so the following optimal results that Sarnac proving his letter I forgot to say what I'm describing here is a it's a result that appeared as far as I know for the first time in in Sarnac's letter which is available on his webpage and in that letter he he basically showed that kappa mu is precisely one meaning that for any large enough q and and almost any element in the finite group there is a lift of it which is our smallest possible side now one can naturally ask why did I this why did I focus on the almost covering exponent why not take the covering exponent which might seem more natural and I guess the the reason is is that I don't know much about the covering exponent but I I'll devote one slide for it and and for the rest of the talk you can whenever I'm talking about the old the optimal almost the and sorry the the optimal small approximation you can also attach to it the question what happened for the covering exponent basically so for the the covering exponent is defined instead of taking the mean over the individual covering exponent you take the maximal ones the worst one in which case kappa the covering exponent is the smallest exponent such that whenever t square goes a little over q to the three times kappa then every box contains a ball and remember the super strong approximation result that we've seen earlier and the logarithmic diameter bound so that bound it actually gives you a a bound on the covering exponent so it's finite and you can bound it using the how good the the graph are expanding but in his letter sarnax actually proved another remarkable result he showed that the covering exponent is precisely four over three and he called us the big holds phenomenon this is similar to the badly approximated element and most elements could be approximated by a lift of norm q to the three over two and this result shows that for that there are some that cannot be so all elements can be approximated by lift of norm q square which is slightly bigger and you can actually construct elements and infinitely many such elements such that you cannot do better and over over q square you cannot decrease the exponent okay there is also this proof of the sarnax also comes with an efficient algorithm to find such a lift and he and he raised the question of can you find such an algorithm for most elements of the of the smaller exponent sorry shy there is a question Igor would you please unmute and ask away okay thank you uh the serum says that the copper is four thirds for any q right does it mean that lim soup can be replaced with just lim a good question no the it's it's lim soup what the sarna constructed and i think it could be generalized to q growing over a prime power of a fixed prime he was able to construct badly approximated element but i don't know if you take q to run say over only the prime numbers can you find such badly approximated and so the lim inf is an open question still as far as i know well but the name that you understand this serum it says for any large q is it for infinitely many q's so for any large q you can you can find the lift whose norm is bounded by q square okay now you cannot reduce this two to one over nine for any large q because i can find the sequence of large q for which the the only lifts go bigger than q to the power of one of one point nine okay thank you sorry for interrupting yeah no no no please interrupt feel free to ask question it's no problem at all thank you okay so this is the this is the the the case of sl2z and now let's see what happened so now let's see what happens for general arithmetic groups and okay so instead of sl2c i'm replacing it with another group scheme i denoted by g and for technical reason let's assume that it's connected almost simple simply connected and i'm going to divide the the groups i consider to two cases the first case might be the more natural case is when the the real points of the group are non-compact and in which case the arithmetic group that i'll consider is gz for example all special linear groups and all split simply groups fall in this setting and the case two is when the real points form a compact group and but there is some for some p the periodic points of the group is non-compact in which case it doesn't make sense to study slow approximation for the arithmetic group gz because gz will just be a finite it's discreet and compact and its size is fixed and the size of the group gz mod q grows to infinity so clearly you cannot have small approximation so you instead of the integers you take the p integers the z one over p and that would be in case two that would be our arithmetic group example of this case i'll say definite inner forms of special unitary groups or definite inner forms of the synthetic group and it is a and and the the strong approximation property which is well understood is is generalized to all of these groups and more and it says that again the modulo q maps are there are natural modulo q maps and they're all subjective for any admissible q for any in case one it's any integer q and in case two any q which is called prime two p okay so the strong approximation uh property holds in in higher in more general arithmetic groups what about the super small approximation well we can again we can do the same kind of construction that we seen before but the second you fix some symmetric generating set of your arithmetic group you can define the cali graphs associated to the finite group gz mod q and sq which is s modulo q and as before strong approximation is equivalent to connectivity and therefore it's natural to state the super small approximation property is the property that these cali graphs are expanders and this is in fact true and it is a theorem and this super strong approximation property for arithmetic groups was it's it's a more common name is called property tau it's a property suggested by Alex Lubotsky which is as the name suggests it's kind of the and it's the love child of Kasdan property p and the selbert 316 theorem extended to other arithmetic groups and there is also a second proof which is the the generalization of the bourguin gambler expansion method i should say that the bourguin gambler expansion method is stronger in some sense and and slightly weaker in another sense it's stronger because it applies for a much larger class of groups not just arithmetic groups but also protein groups proof the expansion property and it's slightly weaker because as far as i'm well that somebody in the audience will correct me it's today it's only known when q is square free but other than that this is so we have two completely different proofs for this super small approximation and exactly as before super small approximation the expansion gives us a logarithmic diameter which will result by the way in a in a bound on the covering exponent but i won't i won't talk about the covering exponent anymore in this talk and okay so the super small approximation is true also what about the optimal small approximation so the same definition for before just okay apply in this setting as well you need to fix some norm maybe the Euclidean norm if your bedroom if your group is linear and the exponent the ratio three over two is now replaced by this normalization constant alpha and you define exactly the same thing you define the individual covering exponent of element in the final group g and gq and kappa mu of q is the average among all the elements in the final group and kappa mu is the lean soup and when q goes to infinity and by the regional principle and the normalization with alpha we get that kappa mu cannot go below one and the conjecture the conjecture properties that uh are these arithmetic groups have the optimal small approximation meaning that they could be lifted any almost any element in the finite group can be lifted in a very efficient way efficient in terms of size not algorithmic okay so now I have some fresh out of the press news and I received this briefly a few days ago that two combining two works the first one by Assing and Bloomer and later on Janine Cumber who removed an hypothesis in Assing and Bloomer's paper this optimal strong approximation results now holds unconditionally for any special linear groups assuming the q are squarely at this point in time it's uh it's I want to add that the way to the stomach prove his stronger optimal strong approximation for lesson two was by appealing to a density uh theorem or the density version of the Stelberg Ramanujan conjecture and this is exactly what Assing and Bloomer and together with the Janine Cumber the gets they get a density theorem that enables them to prove the optimal strong approximation and no other group is known to satisfy the optimal strong approximation as of now okay so and for the rest of the talk I'm going to focus on the second case and the case where the group is compacted infinity and our lattice is the pure arithmetic lattice and and I'm going to show you that in in a parallel field and people working on expander graphs and Lujan complex says they have also discovered a property which is an optimal property which is equivalent to the optimal strong approximation and to describe this property I want to kind of switch the object that we're looking at instead of looking at the Kali graphs which are a bit unnatural because they depend on this auxiliary generating set S there is a more natural complex associated to our to these module q homomorphisms so what you can do is you can take the what it's building of the periodic group and you can question it by the congruent subgroup and these these these are finite simplicial complexes finite arithmetic complexes and analog these are periodic analog locally symmetric spaces obtained by causating by congruent subgroups like modular curves and they actually have a high dimensional non trivial high dimensional structure they they come they have vertices edges triangles and so on so to make this talk more concrete I'm going to look I'm going to associate to them certain directed graphs I'm going just to look at the maximal phases of these complexes this curly x of q which can be identified with a double coset divided from the right by an either always subgroup I which is stabilized over the maximal phase and divided from the left by the congruent subgroup gamma q and this would be the set of vertices of our graphs and we have some freedom how to choose the adjacency operator and hence the adjacency relation and and we have a lot of freedom so for any element in the periodic group you can define an adjacency operator to be the evaporator operator acting by convolution from the right by the double evaporator set II so these are natural geometric operators they are defined on the building and they descend to all the finite quotient of the buildings and clearly there is some a lot of redundant information if I allow you to run over all elements in the periodic group so let me restrict the tension to a specific elements first of all recall that there is a affine board the composition that enables you to write every element in the periodic group is in the following form so the double cosets the representative of the double coset actually belong to the to could be written as a product of this some torsion element and some free element an element in the vile chamber to those you know what it is and I don't want to handle the torsion elements the elements coming I don't want to have anything coming from the finite velcro because they make the computation much more difficult and they also in applications the adjacency operas that we want we want them to have some nice collision free structure behavior on the building so I'm going to take element only from the positive veil chamber okay and and this these are elements that people working on studying the expander graphs and high dimensional expander and they like these operators because these are these are non-backdacking operators whenever you walk on the building you never get back to where you started you only see new vertices as you progress okay so here is a definition it's it's slightly different definition than the usual definition of ramanujan complexes it's it is motivated by the work of lubecki lubecki parts of chev'ski which I would describe in a second and this is we call these arithmetic complexes ramanujan if the corresponding directed graph of them are ramanujan when are they directed graph ramanujan when the spectrum of the adjacency operator of these graphs is contained in the union of these two states the set on the left is usually refers to as a set of trivial eigenvalues these are the eigenvalues in absolute value equal the degree for grass these are usually the degree and minus the degree for directed graph you can have roots of unity times a degree for simplicity let's assume that the only trivial eigenvalues the degree and its eigenvector is the constant function and the second segment could be called the ramanujan and so it's not a segment it's a it's a ball in the complex plane and this is this is the ramanujan part of the spectrum this is the spectrum coming from the the covering directed tree and this is the part of the spectrum this is this is the the good part of the spectrum so to speak and those of you who are familiar with the ramanujan graphs might look at this and be a bit confused because for ramanujan graphs you want that the non-trivial eigenvalues to be bounded by two times square root of the degree minus one but these are for the adjacency operators on the vertices of an undirected ramanujan graph and here we have a directed ramanujan graph where the definition is a bit more a bit simpler to remember just square root of the degree okay so these ramanujan complexes and why what kind of interesting properties do they have so here's what they here's the interesting property that they're satisfied here's their connection with the optimal strong approximation result I mentioned earlier so let's I'm going to fix a which defined the adjacency operator and I'll consider the digraph xq or q goes to infinity and a simpler okay so one can now observe that if you fix some epsilon and you let q go to infinity by a simple union bound if you look at only the vertices in the graph whose distance from some from the origin is bounded by one minus epsilon times log to the base of the degree of the graph the side of the graph then even if it each step you see only new vertices as you progress in the graph this set is grow smaller than the side of the graph the optimal almost diameter property is the fact that if you go a little bit over there over this bound and you switch the mean the minus epsilon to plus epsilon then you see almost all of the graph meaning that the number of vertices whose distance is greater than one plus epsilon times log the side of the graph is negligible if it was zero then this would be the diameter and and we know that this cannot be and this is this does not happen because of the big holds phenomenon so we cannot expect that the right hand side to be zero but we want it to be negligible and this this is what we call the optimal almost diameter and clearly you can you can see the the the relation between the optimal strong approximation and the optimal almost diameter the the two properties are strongly related if you want to be precise and you want to show the equivalence then you need to just pick the norm using the optimal strong approximation result to be compatible with the distance on these directed graphs and now I can state the line of results starting with the breakthrough result of Lubecki and Paris who showed that Ramanujan graphs satisfy the optimal almost diameter this was shown shortly after also by Sardari and this phenomenon was generalized to Ramanujan complexes by Lubecki Lubotsky and Pardynczewski and showing that Ramanujan complexes have the optimal almost diameter which is again you can think of it as the optimal strong approximation result in fact I should say that the the a stronger property was proved namely a cutoff result for underworks and I should also mention that I'm not stating the best result here because I'm fixing this epsilon but you can actually ask for a better understanding of the epsilon and to those of you who know what the cutoff property is and let me just mention that Nasterodi and Sarnak actually proved that for Ramanujan graphs the cutoff is very good the window of the cutoff is a bounded size and there is a question of whether it's true generally speaking in any case Ramanujan implied the optimal strong approximation that's what I I hope you remember from this result now the proof is very simple so I I mean I have heard that every every talk should contain a proof it should also contain a joke but I don't know any joke and so let me show you the the the short proof the short sketch of the proof and quite nice so I'll denote by N the size of the graph and k the degree k and x0 is some of the origin point and for any L S of L will denote the support of the adjacency operator to the power L apply on the indicator function of the vertex this is exactly the sphere of distance L from the origin so on the one hand a simple union bound show that if L is one minus epsilon times log the start of the graph then you cover a small fraction of the graph what happened if L is one plus epsilon times log the side of the graph so let's look at the term in the middle you apply the adjacency operator to the power L on and the projection of the indicator functions to the space orthogonal to the constant function so on the one hand oh what sorry okay so on the one hand if I were to just I can lower bound it by looking at the values of this function at points outside of the sphere and what I get is that and what I get the size of the complement of the sphere times time this quantity which is n to it's n to the two epsilon on the other hand I can bound this from above by four times the operator norm restricted to the non-trivial part of the spectrum and well this is like taking the eigenvalue the maximal eigenvalue of t to the L square right the maximal eigenvalue of t to the A because of the Ramanujan is square root of k and so now we get that the from this we get that the complement of the sphere is also negligible this is the complete sketch of the proof with only one lie not that it's not that it's not true but this is not a trivial estimate it actually uses some representation theory so maybe I should tell you why this is true it would have been true if the graph was undirected and the operator were normal then you can compare directly the norm with the maximal eigenvalue but these operators are not normal not like the HECI operators however these are evahory operators and using the theory of the representation theory of evahory algebra one can actually show that these although they are not normal they are not unilaterally diagonalizable but they are unilaterally equivalent to block diagonal matrices and the blocks are of size of bounded size and that's that's really the the last piece of the puzzle in this book okay so we've seen that the Ramanujan property implies the optimal approximation result and so natural question is are these complexes are these arithmetic complexes Ramanujan if so we get the optimal sum approximation result for the arithmetic group and the answer is that not usually for example most matrix groups are not Ramanujan complexes and the reason is that the naive Ramanujan conjecture breaks down in for most classical matrix group and the first counter example were constructed by Haupy of Tetskish appeal but now we understand better the failure of the Ramanujan conjecture so what do we do since we we we might not be able to prove the Ramanujan so as a so Sarnak and Shwey anticipating this problem already in the 90s they suggested a replacement of the Ramanujan property which says that you don't really need Ramanujan on the nose it's okay that you have some violation of the Ramanujan conjecture as long as there there are few of them and so more explicitly the Sarnak-Shwey density hypothesis requires that for any parameter R between two and infinity and for any q go to infinity the number of eigenvalues okay so here there are some black boxes eigenvalues correspond to irreducible representation of the PIB group with if a whole fixed vector so there is some representation theory in the background and these representation have an invariant called rate of decay this invariant grows when the eigenvalue becomes worse and worse and and so whenever you have eigenvalues which are too large so the set of eigenvalues which are too large is bounded by this explicit bound by the total number of vertices in the graph to the power of two overall okay so for for time consideration I won't explain the exact formula of how to go from the eigenvalue to the to the R of the eigenvalue and let me just describe the the extreme cases and when the eigenvalue is trivial and R is infinity and when the eigenvalue is in the Ramanujan domain then R is two and note that the the Sarnak-Shwey density hypothesis an easy way to remember it is that it is a linear interpolation between the two exponents when R is two and infinity you see that for infinity the left hand side is of finite size independent of q and when R is equal to two then the bound is trivially true and so this is the last slide and I want to I'm I'm I want to mention some two ongoing projects that I'm involved I I'm not stating any explicit results and just mentioning that this that in both of these projects we have a strategy to prove the Sarnak-Shwey density hypothesis and the strategy uses two main tools coming from the Lengens program the first is the endoscopic classification of automatic representation for classical groups this was done by Rogowski completely for the unitary three by three matrices and by Arthur for quasi-split classical groups and there are some follow-up improvements for inner forms of classical groups by Taibi and Kalleta and Mingez and Sheen and White and the second key result that we need is the generalized Ramanujan-Peterson conjecture which is now a theorem and it's due to a long list of famous mathematicians and their works on the Lengens program the Ramanujan conjecture starting with Eichler, Shimura, Deline, Lenglens, Lozel, Sheen and the list goes on and on I I didn't give credit to all the people that contribute to the final theorem that enable that prove the Ramanujan-Peterson conjecture and I'm not even stating the Ramanujan-Peterson conjecture explicitly and the point I'm trying to make is that the the Sarnak-Shwey density hypothesis can actually be deduced from a combination of the endoscopic classification and the Ramanujan-Peterson conjecture and some perhaps other working hypothesis coming from the Lenglens program and in the two joint ongoing works the first one on the on definite inner forms of U3 we actually prove their prongerism that the Sarnak-Shwey density hypothesis oh I should have to say this is a joint work with Brooke Fegon and Kathleen Neymar-Witchett and Oripatmanchevsky we show that for the unitary 3x3 matrices not only do we have the Sarnak-Shwey density hypothesis we actually have Ramanujan complexes which is surprising but we we get and in particular we get the Sarnak the the optimal sum approximation result and in another ongoing joint work with Matilda Gerbelli-Gothier and Henry Gustafson which I should apologize to because in the abstract I sent you there is a U in in his name it should be an O and so in in that joint work we are also considering the case of definite inner forms of the autogonal special autogonal group on five of five matrices or if you want the the projective symplectic group of four by four matrices and these are and this is the the the the two results and hopefully the beginning of the understanding of how the language program could deduce the Sarnak-Shwey density hypothesis and eventually gives you also optimal sum approximation approximation results and optimal almost diameter results for arithmetic complexes and I think I'll stop here thank you very much for the invitation