Andrei Shubin: Circle coverings driven by arithmetic sequences (NTWS 299)

Thanks a lot for the invitation. I'm really glad to give a talk here. Uh yeah, so I will talk about a joint work together with few more people also from gratu and it's a recent work. has been on archive since like 5 months and uh okay let me see uh so in this work we solve some problems which can broadly be speaking like broadly speaking can be called like soal covering but um depending on your fault you can call it differently you can for example call it u if you do diantine approximation you can say twisted dontin approximation with restricted denominator S if you do number theory you can maybe think of local scale distribution model of one if you like fractal geometry you can think of limb subsets uh so the plan is roughly this I will first give the original motivation coming from probability and tell you what's the what is called random circle covering then I'll talk about some modern extension and um tell us what we're interested in then I'll spend some time on some proof ideas and if I have time I'll also mention some applications. So okay and please feel free to interrupt me or ask questions in the chat. So the original circle covering problem is called Vetski problem. It's named after Arin Vetski who posted it first in his paper in 1956. And the problem is as follows. uh take unit circle or unit interval and then try to cover it by a sequence of random shrinking arcs and uh infinite covering means that every point on the circle will be infinitely many of those arcs and almost surely is necessary here because it's um a probabilistic statement. So potentially it can happen that they all have the same center but generically we expect that if those arcs are large enough they will cover the full circle infinitely many times. So the question of dinski what's the condition on this shrinking sequence of sizes so that uh the random marks will cover it infinitely often. I think this problem is quite well known in probability but maybe it's not very well known in number theory. That was my impression. Uh you can also look at it from a different angle. You can think of it as of random analog of dish approximation theorem. In the usual dish theorem we approximate numbers by rational numbers. But in this case we approximate them by random points. And then the question is what's the um approximation rate so that um the analog of DS theorem holds almost true. That's exactly the same question just written differently. Okay. So let's gain some intuition and first look at an easier problem. What if I want to cover a fixed point for example zero infinitely often? uh well in this case the answer is given uh simply by application of B can contain lema. So since they're independent by the second lema, it's enough to have the divergence sum. And on the opposite side, if the sum is convergent, the first barrel container lema tells that that the probability of infinite covering is zero. And in fact, you can extend it slightly more. You can also apply the same to show that uh you can cover almost all points on the circle with respect to the big measure say. So uh this is just a slightly more involved application of B canalis lema. But it turns out that the Vetski problem is actually harder because there you are not allowed to have exceptions. If you want to cover the full circle uh having divergent series is not enough. There is a slightly maybe easier way to see it. Suppose that my um sequences of the form 1 / n square. So this is convergent. And even if I place the arcs in the most efficient way just right next to each other. So it's one then one quarter and It's a bit disproportional but you see the total total size is finite. So of course one cannot cover something of positive measure with just a finite finitely longer. So we immediately see that divergence is necessary. But if you take something like this one over n login and a priori it's not clear if it's enough. So it took quite some time to solve this problem actually about 16 years from the original paper. Uh the first progress was made by detski himself. He showed that in fact this rate is not sufficient. It's not a covering case. But if you take some somewhat larger rate this will be sufficient. And then there was a number of works kind of squeezing this limit to a correct threshold by either proving an upper bound. So given sufficient condition or proving the lower bound by giving necessary condition for example already the next portion of works by beer than kahan revealed that the correct threshold should be at around 1 / n and it was open for quite some time. For example, Erdog claimed at some point that it's um it is a covering case, but he had never published the proof. Instead, he published it as an open problem in one of his papers. Then later, it was shown by AI that it is actually a covering case. And then when Pro also showed that even slightly below the rate below that is also covering. But somehow it was not yet the answer to the problem because um you could always find a rate somewhere in between those upper and lower bound. You could for example take something like one over say n minus one / n login and this is below the boundary of underro but it's still but it's above the lower bound of [snorts] and kahan okay and so finally the answer was given it was solved by larep and I think this is also something which is well known in probability it may be not a number theory. Uh so he gave the condition which was sufficient and necessary at the same time. So a criterion which you can see here and it's actually quite simple. Now as soon as you have a shrinking sequence of sizes and you can compute the sum here you can determine whether it's covering or non-coing case. So for example [snorts] if I again take the rate 1 / n - 1 / n login I believe if I haven't made the mistake I should be covering this. What if I take something just a little bit below this 1 / n - 1 / n login and say put a a power 1 - epsylon here for arbitrarily small positive epsylon I should be getting non covering And so this way you can it's like it's fully settled. And uh so one more thing I want to say about it. So it's kind of interesting that this problem it's uh it was foundational for this theory which is now called multiplicative chaos and used a lot in the modern number theory especially in connection with um studying the size of za function on a critical line or studying the sums of multiplicative functions. Somehow it was developed by Kahan who was actually very interested in discovering problems. He wrote quite a bit of quite a lot of works on this generalization of SH into different directions like different metric spaces, different dimensions, different probability distributions and in particular he developed okay he developed this theory which is called multiplicative chaos and he gave a different proof of SH's theorem using his theory I think it can be found in the book is it it's either a book of a survey this random multiplications, random coverings and multiplicative chaos and uh I cannot tell you too much about multiplicative chaos but um I can give you roughly the basic idea. So the basic idea you uh construct like a sequence of random measures and you look at the limiting behavior of this sequence. So in the covering case what he constructed is simply the indicator function of uh the complement of that random mark. So if the point survives covering it will be supported by uh such single measure normalized to be like one and then if the point survives covering by say first many arcs it will survive the product of measures and then somehow there is like a threshold if um the chef's criterion give you divergence So the arcs are large enough this limit will be something degenerate I guess it will be zero if it's um and otherwise maybe you can say something about the structure of uncovered set like uh something about it structure for example and somehow similar point of view is used by um in the works studying the exita function for example the one by Saxman and web or Instead like the random measure is given by the oiler factor and the function is approximated by oiler product and you have some [snorts] uh and you have kind of similar type of um uh like you also need to analyze the limiting behavior of this product and depending on the movement you may have something called uh critical Gaussian chaos. it can be super critical or subcritical and uh so I find it kind of interesting that this whole theory kind of emerged out of um the interest to discovering problem which is very classical. Uh okay so maybe I should stop for a second and ask are there any questions? Okay then if not let me uh let me continue. So now let me tell you what are the modern extension of this problem coming from the funing approximation and what we were interested in. So you may ask what's the if there are any deterministic analoges of discovering result and in fact uh there are probably not so many the most like obvious one is dish theorem itself it can be interpreted as a covering result where random points are replaced by rational numbers well I guess except for for rational numbers themselves they're not covered but it's a countable set so it's not really it's not a big deal. And then you have um yeah we have covering with uh centers at refractions and arc sizes given by whatever the right hand side here. And what you can do you can uh test the list theorem against criterion. You can pretend that this is uh an outcome of your random experiment and um put it into criterion. So we just need to rename um things like go from Q to N by ordering V fractions so that the arc sizes is not increasing and if I didn't make a mistake here I think you should be getting something like this zero something below one so this is below the threshold in this rep which was about one over size uh so generically We would expect that the covering should not hold but it holds nevertheless. And it holds because in fact rational numbers they are not that random. In fact they they have some structure they dwell spaced and on average they have better covering properties than uh a typical random outcome. So in that sense G theorem is better than its random analog. Um a similar example of this is um also a well-known sequence which is called chronic air sequence. This is just a sequence of fractional parts of irrational delays. And it's well known for example that if alpha is rational this sequence is uniformly distributed modulo one. But this is just like a global scale behavior. On a very local scale, the sequence doesn't look random at all. It's actually also very structured. There is for example theorem saying that there are at most three different distances between um say first end points of the sequence put on the circle. It's called the three gap theorem. And uh like infinitely often those gaps they're roughly of the same size. So it's actually also has quite it's a good candidate for covering and um here we have this result showing that the threshold is at also more than twice better than the threshold in the random case. I guess this should be attributed to hing. Okay. But then apart of from that I cannot really think of any natural examples in number theory. a lot of interesting sequences they come out from uh as subsequences of chronicer sequence. So I can think something arithmetically interesting like squares or how higher powers of primes and study such sequences for whatever choice of alpha you have. But those questions they're quite hard usually here you would expect that this is more random but showing something like covering at the correct um correct scale of the arc size this is very hard and probably out of reach. Um it's quite hard because even an easier problem it reduces to covering just zero infinitely often it's u uh it's also quite hard. So I guess to the best of my knowledge these two results are still the best for squares and primes. Um so this old work of Garesco given two hertz for squares and more recent work of Matamaki given the result for primes but even improving this exponent is like very hard. getting the conjectural order is like probably out of reach. Well, at least for for humans. Um, okay. So, yes. So, since it's quite difficult, the usual simplification everyone does is uh by sometimes does is by looking at u instead of fixed alpha looking at typical value of alpha. So you can ask the same question for almost all values of alpha um sampled from an interesting measure with respect to an interesting measure and um then many such problems are called metric. So I thought calling it like metric covering make some sense and in this setting it's again a problem about random covering but there is an important difference from the original question note that the centers they not they're no longer u independent. So as soon as you know something about the location of this point a little uh as soon as you know something about this point it gives you some restriction on the location of the next one and this uh dependence it um kind of diminishes the quicker the sequence grows. So for very quickly growing sequences this is pretty much uh as in the um independent case and the hardest cases is when this is something slow. So um so now we can basically state uh like a general problem which is studied quite a lot in different um context. So given this interesting measure whatever it is uh try to say something about the size of the covered set. So it's the size of all those numbers which are infinitely often approximated at a certain rate and um here I can say there are few directions you can push it to. So I can basically uh see at least four directions here. So one direction you can for example if your arc sizes allows you to cover the full circle try to prove that it's a full circle and um I'll call it direction number one. uh another question I guess maybe you know that it's a full circle or whatever it is and then try to get the optimal rate optimal approximation rate you need for this like try to get as close as possible to the threshold I'll call it direction number two uh another direction you can also try to get the result for as many different sequences as you can especially those which grow slow. I'll call it direction number three. And one more is uh by um changing the measure like usually it's measure but maybe the same result can apply for smart measure on alpha which sometimes has interesting applications. So at least we have this like four directions and uh kind of there are number of works which push us into the different pushes the different direction and getting the precise statement here. So we were mostly interested in the direction I guess number one and two we were trying to show the full social covering with this dependent centers and somehow interestingly we couldn't find a lot of works in existing literature which actually does that. I might I may not be aware of some works or maybe I'm not like citing something in this case I'm sorry but um kind of one notable exception we found which did um establish the sole covering is this earlier work of funing and tit they showed that this circle covering holes so that the covers have different interal for um for this special sequence to the DN which is called sometimes doubling mop. The paper is like dynamical mostly and uh this is just one of the results and as far as I one can tell from this result it follows for so the precision rate in this case one has to remove excitement here so it's in that sense it's not very precise uh so you need to have some margin but I think the strongest point of this is that they could replace big measure by what is called Gibs measure with some very wide class of fractal measures which I can't tell you a lot about but it's like direction four is maybe like the strongest point of this result and in another direction recently there's been also some works first by Christensen and person then it was improved by how Ramirez uh which gives like an almost covering. So instead of full circle we get that we got that liic measure of the covered set is full. So in that sense it's less precise. But then they got well the optimal rate for that which is I guess something like the usual hing threshold. If it's divergent, you have um almost all covering and it's very strong in terms of the sequence. It's actually applicable to any increasing sequence of integers and uh I guess it was also applicable to some other measures with what is called FIA positive FIA dimension which like essentially means that you have polomial free ADK and u but then maybe you have to relax some restrictions on the sequence or on the rate this I don't quite remember but this is like paper which pushes the same problem but in a different direction um okay then there are some other relevant works which are about studying the gap distribution between the points. So the problems about gap distribution they was somehow [snorts] getting popular after works of Rutnik Sarnak and Zaharesu like 25 30 years ago and uh those problems they you know to be hard even in the metric sense but for like quick sequences we more or less know how to do it now could get the limiting distribution of consecutive gaps between the points modulo one. But there is this um kind of extreme version of this problem where you study adult layers either like very large gaps or very small ones like in particular you can study the maximum gap. So if you put end points on the circle from whatever distribution you have an average gap will be of course of size one / n. But if it's random then you expect that there will be some gap which will be larger and uh the random uh model predicts that this should be of size uh roughly login times larger than the average size. This is um some old working probability due to deo and recently it's been generalized to uh um sequences linary delays of um integer sequences which are not independent. But the slary sequences they are they kind of known to be a threshold uh between kind of what can usually be done and what is maybe very hard to do and uh these are all the sequences which grow at least geometrically fast. So something like 2D todn for example. So basically if this ratio is bounded away from one sequence is lucky and uh there's been a series of work getting the bounds from those maximal gaps the first one by Sam Chow and Nicholas Techno who get the bound which was just two logs away from the conjectured one then Edward Stefanes who improved it to the second power and very recently like two or three months ago uh in the work of Paris and Young uh they got um they essentially settled the problem by getting both upper and lower bounds of the with the explicit constants and um why am I mentioning this? So this maximum gap problem is directly related to the arc size in the covering problem because you can use the size of maximum gap to deduce the covering with the size of the maximum gap. If you know that your maximum gap is of this size, you know that every point on the circle it belongs to some gap which is of size at most maximum. So you know that every point is at least like at most that size close to the random point and so you can you can just get covering result out of the out of knowing this gap size and the point is this this is of course that's a different problem which is of independent interest but in this case it gives something a bit stronger than what you need because it gives you a simultaneous approximation. So it's kind of not a surprise that you have an additional log here uh because it's simultaneous. But what you can deduce you can directly deduce that you have covering at uh the rate given by the size of the gap. So in this case this is the red this was what I call direction two and it gives you the full circle and uh from this works it was also that you can put any interary sequence here and uh I guess in Paris Yang's uh work it was done for the big measure but in two previous works um due to some application uh they also considered this year polinomial decay measures. I'll maybe talk a little bit about it at the end. Okay. And so kind of what is our main result? We were directly interested in this covering and trying to get the optimal um arc size and we've managed to do it. So what we did show that the circle covering holds uh for any real validary sequence but what is more important that it holds at the correct scale it's only it's only like the constant side of the away from the conjecture to so this constant C here in the cover. It depends on the sequence we take and we didn't try to optimize it but this is like something like 1,000 maybe and um then the covering holds fully big almost to alpha. This is like this is the upper bound. But then a bit later we were also interested in seeing if we can show that it's optimal uh by getting the matching lower bound. And we actually managed to prove the lower bound as well. But um somehow the technical part of the proof was quite different in this case and it had different uh bottlenecks different restrictions. So somehow interestingly we got slightly different uh outcome. So we could relax for example in this case we could relax the restriction in the sequence. It doesn't have to be luck anymore but it can be much slower. But then we couldn't do anything other than li measure. In the first case we could extend it to the polom to this like measures. In the second case we could do it only the sequence like extremely fast and otherwise it was only big measure. Uh but um somehow as the main outcome we could show the essentially the analog of ditki covering in this um metric sense. And um well I guess the further direction is to try to reduce um the rate growth of the sequence in this case. Uh okay maybe I should stop again and ask are there any questions? Okay. So, um in the next part I guess in the last large part I'll try to give you some insight from the proof. Um so the main idea of the proof is to model the point process on a circle by uh a branching process on a binary tree. So uh one can see it as some sort of branching random work in uh in probability and discrete math. I think it's called uh percolation. Uh but we call it uh um tree coloring. I think after all it's kind of all the same things. Um the idea is the follows. So every time I put a random point on the circle, I put a point on the tree. uh by coloring one of the vertices and um let's say I want to model the arcs of size roughly one / n. What I do? I put the first point on the top uh top level where it's just the full interval and color it. Then I go to the second level and look at the next two points, second and third. I put it onto the tree and I look where they go. If they both went into the first half of the interval, then I color it. If the second interval remained free of points, I keep it uncolored. And then I continue this way. Then I add next four points. I see where they go. I color vertices depending on the location of the points. And uh so in this case, coloring depends on really where points go and when do they come there. So if I want uh if I want to prove covering, I want to um kind of color my tree aggressively by adding a lot of points. So if I want to model large arcs, let's say 1,000 / n. Okay, something like 1,000 / n. I'll add 1,000 points already on the first level. Then I add the next 2,000 points on the second level. than 4,000 and so on. So my team will be highly colored and then you can expect that with the high probability um okay I guess I go a little bit ahead of myself I should first um uh say what does coloring mean. So every number delta which is a number on the unit circle it has a unique binary expansion right so it can be written as a a sequence of ones and zeros and apart from rationals this expansion is unique for rationals we can agree to choose one but what it means it just means that we can represent every number by a sequence of nested diadic intervals so we can just look at number delta as as if it was a way down the tree along the nested diic intervals and so such each number is like a path and what does it mean that the number is infinitely often close to a random number it roughly speaking means that every number shares the same diic interval with the random point infinitely often. So it means that infinitely often each such infinite path would have a color vertex. So that's the idea. If I want to show that there exist an exceptional point which is eventually not covered by a random arc. I look for a path which is eventually free of colored vertices. So I look for something like that. And then the idea is I don't change the tree itself. Every diic interval on the tree has a fixed size with respect to the level number. It's 1 / 2 to the k. But I can change how many points I add to the tree. I can change the constant here. So that this is roughly c / n. And now if I want to show coloring, I want to show um a lot of coloring. So I put this constant to be large. I take it maybe a th00and and then I add a lot of points. then very likely I won't find the path like this of the second type. If I want to show non- covering I will take C to be very small and I will be adding as like a small number of points and then the tree will be highly uncolored and with the high probability I maybe find such path and that's the idea. So somehow we get this one to one correspondence between uh um realization of my random sequence and the colored binary tree by just reducing it into two cases. For non-coing we look for um non-coed path. For covering we prove that every path is colored. And it's kind of much more pleasant to work in the framework of in this framework because now you can for example um recover you can improve the first result of Deti by just doing some pretty standard probability and combinatorics on the tree uh [snorts] by getting the correct orders of upper and lower bounds. Well, in our case since uh the centers are dependent, we had to be a bit more careful and do a bit more work here and we actually needed to use u some free analysis. So first I'll tell you how does the upper bound go. So in this case we want to show that we cannot find a path which is uh infinite and uncolored. uh but okay technically it's quite hard or it's not very clear how to work with an infinite object like this. So what we did instead we approximated this bad event of having such a path by a sequence of slightly better events which is still highly unlikely events that we find a lot of um very long finite segments of uncolored path and we can arrange them to be arbitrarily sparse. It's because the main bed event would imply the uh the existence of such segments. So the goal is to show that the probability of each such event is small. So something [snorts] like say 1 to the 2 to the ra where r i is the height of uh segment and then the same here. So if you show something like this this bounds are summable and by bal can tell we will get that okay there is probability one we won't see such pattern any such pattern and then it implies of course that we cannot have an infinite colored path that's the main idea and now we we can work with this finite segments more efficiently we can actually apply analysis so I would say free analysis part here is quite standard for [snorts] um any like analytic numbers who um who usually like to compute movements. Um in this case we like as usual need to compute some movements of trated fully series. Uh so this is I would say something quite standard. If you have an interval, you can replace the indicator function by some smooth indicator which has a a f series quickly decaying after some point. So you have I don't know something of this shape and then the point is um the point is that we need to compute mments we need to evaluate mments of such um exponential sums applying chambers we find that okay the probability of having a long uh uncolored segment is just a second moment of a very long product and this is the crucial place where we need lunarity. We need to have sufficient independence of different sums since in general when you have such a high movement it's quite hard to evaluate it but if you have second movement it's much easier and lunularity gives us some enough independence to say that okay [clears throat] this long second movement of the long product is just a product of many second movements. So somehow roughly the points coming on the different levels they are they have indices different like two times roughly speaking and because it's likeary this would be a very huge ratio like if you go back to the original index might be something like n to the 100 and then roughly this sums independent of each other and uh one technical way to see it is just like by orthogonality essentially you see that you need to count um how many how often the the total faces close to zero and here you essentially have the diagonal contribution only because different u differences there at the very different scales and they kind of don't see each other. So this counting can be done then in some inductive way and okay the idea is then we get this bound. We had to choose this r to be large because after all there will be some union bound over u the number of possible finite segments between two levels. So we would need to compensate for something like that. And so that's why we need to compute this very high moment but then for sequence it works out and we get what we want to get. So that's kind of the idea for the upper bound. For the lower bound we work within the same framework but uh the approach is slightly different. So in this case in this case we don't color the tree too much and so we want to look for uh for an infinite path which we can kind of get by uh modeling a random walk on the tree. So since we don't color it a lot, we with a high probability we could find a cluster of vertices somewhere on the top of the tree which is not colored. Then with a high again with the high probability on the next step those vertices they will produce u the next set of vertices or the next level which also won't get colored. And we continue this way at some point. Okay, we get sufficiently large cluster. So maybe some points will get colored. But the point is with the high probability it would be just a tiny amount. So maybe one point get colored we ignore it and our cluster get slightly smaller but on average it grows. And we continue this way. Every time something got colored, we kill the whole branch. Then we kill branches here and we move on with whatever cluster remains. And um this is of course a random walk in the sense that you look at how many potentially infinite path on the tree you can have. So every time the number of such potential ways increases we do the random box does the step well in the right direction and in unlikely event that too many vertices gets colored we do a step to the wrong direction but if we don't color a lot then this probabilities would tell us that the likely the random work would go to the right direction and will never come back and so technically it's done by computing by evaluating the conditional second movement. So we kind of assumed that on the previous k minus one level we only had good events and say less than 25% of the points in the given cluster was not colored then we assume that on the case level finally the bad event happened and too many vertices get colored. So we need to evaluate how um how unlikely this is by computing the second moment. So we again use chashef and um evaluate some probability and after all it reduces again to some kind of moment computation though well technically it will be a bit different we would have like sum over intervals and because it's conditional we also integrate over um a weird set. So it's like the set obtained from knowledge of the location of previous points. So it's the union of like tiny intervals and that's what creates some restrictions on uh which measure we use. So like in libe case it works out but in more tricky cases it actually um kind of gets more complicated and we had to assume like very um we need to take a high rate of growth very sequences. So this is kind of the idea like after all of course the point was to show that this is something summable and then by can we get that u the unlikely event won't happen infinitely often almost surely so we would get that probability that uh the good event happens all the time after some point it will be like bounded away from zero and then because the tree is infinite by So I don't know if infinity lema we conclude that uh we will find such a path almost truly that's the idea and um okay that's uh I guess that's all that I want to say about the proof and in the remaining couple of minutes let me also mention some applications and extensions of that. So um there are a couple of things uh the first one is not really an application it's more like the extension of the same method on a slightly different problem. So uh there is this opposite regime when instead of the soal covering you have um very small arcs and what you can cover is only like a tiny proportion of the circle. It doesn't really make sense to talk about an measure of the cover set, but it makes sense to talk about its house or dimension and you don't need to know what it is. It's just like a good indication of the size of the set like uh next level. And um in this problem for like uh those uh um restricted denominators, it's u already kind of done in the previous works. But um there is this more subtle direction of looking at the intersection of random limb subsets and non-random sets. Like if you have two tiny sets of u some host of dimension below one um you can expect that the intersection would have an even smaller u host of dimension. This question they the motivation comes from some problems about like approximating um numbers with missing digits by rational numbers or numbers of missing digits. Those are the examples of the fractal set. So for example, a well-known example is a counter set counter of middle third set which is a set of all numbers which don't have digit one in three-digit expansion and it has house dimension something like look two or look three. So then you have for example a fixed set and a random limb subset. uh the kind of the prediction for what the size should be was made uh for example in this work um by Guju and Duran who modeled rational by random points and they kind of showed this formula for um in a really random case and also in the uh case when in this like uh fraction of delays case but then but but for the sequences which are very fast. So because of whatever methods they use, they needed a lot of independence and they could take I think something like end to do maybe 100 and or even faster and [clears throat] as um an application of our percolation approach, we we were kind of able to translate it into the language of the tree. We could extend it by quite far by getting u like lonary sequences that's real valued and in fact we could also do some arithmetic arithmetically interesting um slow cases slow growing sequences like I think power monomials powers of primes and polomial values by reducing it to what is called GCD sums. So and uh because at the scale of house door dimension we had some it's a bit more flexible we also could kind of through GCD sums we could cover even like very slow sequences in this case and another thing is uh like a direct application it's related to this famous little conjecture which tells you that for any pair of real numbers you have a certain um um like [snorts] it would get close to zero at a certain rate and this is a big open problem but it's known for example that uh the only interesting pairs you need to look for are those which are badly approximatable numbers so all alphas and betas which are far from rational and u kind of there have been a number of important works towards this uh conjecture uh for example one byton In fani I think they uh kind of showed that even among this badly approximatable pairs the proportion of numbers for which this conjecture holds is quite large by using a measure like a special measure supported on badly approximable numbers and this measure has polomial fiery decay that's why it was kind of important to mention it during this talk and uh there are some variations of this problem one of them was introduced by Hus Jansen and Christensen some years ago. In this variation, the second factor is shifted and the shift has to be uniform. So you ask for the best rate uh uniforming the shift in the second factor and then this problem is really about covering um and so that's why our results they replicable here. So I guess the first result of first work of highness Jensen and Christensen themselves uh was using the bounds on the discrepancy of the points modul one discrepancy it's it's kind of sensitive to any violation of order you can have. So it's actually quite large in a random case it's something like one divided by square root of number of points and in this case it leads leaded to the bound like square root of login instead of this question mark. Then there was this idea of uh Simon Nicholas on uh bounding instead just maximum gaps and maximum gaps much smaller because those are just gaps and discrepancy it's kind of sensitive to all the scales and even clustering or any violation of order but gaps they are like naturally smaller and they replaced this square root of loop by a double block and And then the power was dependent on the bound and it was improved by Edward. And uh uh so as an application of our main result we could since we kind of get the direct bound for the covering we could replace this by just a constant C. So kind of removed everything from the numerator. And uh [snorts] the last thing I say uh what we expect to have as optimal uh bound here is probably one more double log in the denominator roughly [snorts] speaking. Maybe it can also be multiplied by like triple log and more logs. uh and something like this can be obtained if one can extend our result from linary sequences to all integer sequences which u which is more challenging but I think if you can do it then one one should be able to put another double look here and okay I guess that's all I want to say uh thank you for your attention