 Thank you Stefano for the introduction and also for the invitations. Thanks also Jean-Christophe for the invitation. So what I'm going to talk about is on non-conventional Ergolic averages and this is a work done partly with a Rio Moore who is my PIG student and also somewhere where before another PIG student I had was David Duncan but also going to be mentioned there. Okay so let's introduce the problem first. This has to do with the Erdog student conjecture which is from 1936 which is to say that if you have a sequence of integers such that no the sum of the reciprocal is infinite then this set must contain arbitrary long arithmetic progressions. So this conjecture is still open so for the graduate student might be interested in working on it it's still open okay. Now I want to just to talk about now some of the some of the result in this direction and also what was the motivations and the work by the PIG student on that. So first try to understand conjecture is that sometimes these conjecture are made and they are necessary and sufficient conditions but in this case the fact that the sum of the reciprocal is infinite is just the sufficient conditions in terms of these conjectures. If you look for instance as the example which is there when you take the sequence 110 1100 101 102 1000 you see when you go to a powers of 10 they increase the size of the arithmetic progressions then you get a series which is summable which is not summable which is infinite which is summable sorry and therefore you have arithmetic progressions but no the sum of the reciprocal is finite. So this is just a sufficient conditions but it attracted a lot of interest so just for convenience the notation which you use n in bracket like this to denote normal set of integers from 1 to capital N. Okay so definition of a set a subset of natural number with positive upper density if you take now the cardinality of a intersection of the lines the segment 1 to n and you divide by n you take a Limb-Soup of this ratio then you get something which is bounded below by a constant which is alpha which is strictly positive. So one important partial result on this conjecture was the same as the famous Meredith theorem which is to say that if you take a set of positive upper density the set of a positive upper density is not too difficult to show that if you have a set of positive upper density then the sum of reciprocal of integers which belongs to this set is infinite the small exercise is not difficult to prove. So in some ways if you take a condition saying that the set of positive upper density then you have automatically the assumption of the Erdos-Turent conjecture to be fulfilled. Now this theorem was proved by Meredith basically by hand this was you know to the first to try to get it just for the case so I need progression of length 3 then at the back arbitrary no progression of length no k. Now if you look at the set of prime numbers the set of prime numbers also satisfies the Erdos-Turent assumptions in a sense that I even went to for the talk and just for you Stefano went and looked to know how far can I go back I went back to 1737 you see so that you know that I came no prepare and doing that seriously okay so you can even find a lower bound for that the sum of reciprocal it is clear the order of log log n plus 1 minus log pi square over 6 okay don't ask me to prove that I'm not going to give you a proof of that now but it's not by a fact that this is true and this tells you that no the sum of reciprocal also the sum is infinite. Now why am I mentioning that is because we know that the set of prime numbers contain arbitrary long arithmetic progressions this is a result done by the green and tau but beyond these two which is basically a set of positive density and a set of prime numbers the Erdos-Turent conjecture is still open okay now in 1977 first and their proof by using a given another proof of Smedi-Ferns by using algorithm theory okay so it shows that in fact if I take any major probability measurement system and a set E with positive measure then this limf is strictly positive and the positivity of this limf is directly connected to the fact that for an appropriate dynamical system then if you take your sequence you can embed it into this dynamical system and show that in fact the fact that it has positive upper density guarantee the existence of an variant measure which makes this dynamic this system basically know a measure presenting system okay so if you want I'll give you a little bit more detail a bit later about it how you make a connection between the set of integers and the dynamical system is there a question there is no question okay so that's what you have and so that's provided a quite now famous proof of this very different by using no algorithm theory yes so you see you see what you are doing here is that you have taking the limf okay of one of the end the sum from n equal zero I think I use zero to n minus one and then you have new of the intersections from I equals zero to k minus one and then you have s minus I n of e right okay so if you do that you integrate that what you have is one of the n integral of a sum from n equals zero to n minus one and then what you have here is know the product from I equals zero to k minus one of the characteristic function of ease that you apply to s I n that's what you integrate respect to a measure new so this quantity here when you move the integral there this is equal to one of the end the sum from n equals zero to n minus one of the integral of the product from I equals zero to k minus one of the characteristic function of e applied to s I n d new this term here is exactly equal to this one agree yes so the difference with a thing for instance if you gave the case k called k called three what you have here is a democratic function of e that you take at s n of e the average of that are the classical erratic averages but you multiply this function by the car this function of e that you apply s to n e s to s n of x and s is it okay and that's what you integrate respect to new and that's what gives you this because the product of a car to six functions the integral of the product of a car to function is the same as integral of it as a measure of intersection of the two set yes okay so that's what you have here okay so then if you want to you see if you see s what you have here in terms of the action we have the action of s and s square right but s and s square in fact are commuting transformations so the generalization that came later on of this result by first and very castle son was to use s1 and s2 where s1 and s2 are just commuting transformation question is to do can you understand a bit better this structure of this set of positive density okay which great know all these arithmetic progressions and all that now if you follow carefully the proof the connection between this set of positive and potential I said it's a dynamical system and the dynamical system the fact that we have a positive upper density guarantee the existence of an indian measure and this indian measure is greater directly to the the constant alpha that you have here so there's another also reason why I'm looking at that is there's another proof of your stand there which is also to prove that we have infinitely many primes which is a proof which is dated from 1955 and again the basis of that is also is a topological proof which is using also the set of arithmetic progressions they concede the collection of all the arithmetic progressions this arithmetic projection constituted bases and with these bases you can express proof by contradiction but you cannot have no finally many primes so so how can you do get some information one way is to put a weight in front of that put a cn in front of this and see if no by any chance what are the condition on cn so that will guarantee that this is going to be strictly positive so before getting a deterministic weight you may try to find the random weight and this random weight can be a sequence of 0 and 1 coming for the characteristic function of the set a of positive measure and a transformation t which is measure preserving is it a new idea not really because this idea where you have to take averages like that so if you take the simple case one of the end the sum from any cold zero to n minus one the characteristic function of a t and x times no g I take this just for the case k equal one yes and compose is that this type of things the time phenomena where you have no a sequence of zero one when you get one you get one when t and x belongs to a so if you look at the set a here you are in your capital set x this is your set a you have your map which start from x here and then you go back t max may be here t to square here in here then you go here after t and x then you may stay here for a while but you get out and then you come back here again no it's not that you hit your set a you are going to get a sequence of zero and one so t and x is equal to one if t and x belongs to a n equal to zero if not so basically you are looking at the average is that once you fix your x this is the same way that you are going to apply to all the dynamical system that are coming on the right hand side so what does this mean is that in fact what you are doing is you are averaging along the sequence of return times to a set a and so this is called no return times phenomena this concept is not new the first one to have to think of thinking about getting some kind of weighted averages where the weight is going randomly by the return time so set is as far as I could go back to Antoine Brunel in his PhD thesis I think is 1966 then in 1969 he published a paper by Michael Keane where this idea is amplified so Taylor was explaining to explain you have a sequence of zero and one we look at the subsequence nj x over multiple recurrent averages where nj is the jth return times to a set a of t and x to a so the previous question that I was raising here is that would be a cabling asking whether this has a limit which is frequently positive okay now some work has been done on that so try to know what was done before I've been obsessed with something the study of return times as I said by Brunel is his thesis 1966 that's also another answer to a question the Stefan a little bit you get the integral of that that's what about trying to explain with a case k equals 3 right okay so basically the question is put some weight here and try to understand or try to see this limit exist in weekly in norm almost everywhere so let's look at no what was done in this case if you just take a weight equal to one the study of his averages has been a no a study extensively from non-conversions so who are both the case s I equal to the power of s this was done by Austin crime 2005 okay the case s I commuting was done by Terry Tau in 2008 the case where s I generate a Neil Potten group which is the most general case that you can think of was done by a Miguel watch Miguel did it when he was a graduate student so that's you don't want bring this issue here saying that maybe a graduate student can solve your daughter and conjecture am I wrong so it's doable okay and really what was impressive by what Miguel did was the fact that he used basically a very simple tool is a hand banner there and basically his arguments is hand banner but well no the elders more senior while looking at more powerful tools and all that he came up with a simple idea and got the most general case possible if you go to if you try to get something a little more like an important group of solvable group then they are contra example to this okay for the point was conversions well the point was conversions for that is a problem I've been studying from quite some times and some time I think I have it some time I don't so it happens okay but I keep working on it and making a time to try to get in general but the general which is known and be correct and was done only for the case k equal to by jamborega in 1990 okay so now let's see what we did okay so first we have to define a more general process what is a process so you take a non-negative integers you get xn which is a bound and a measurable function on some probability measure space that's what I call a process then one can we say that this project is going to generate a good universal weight for a process which could be point was in norm if for every probability measure preserving system space for which the process is defined these are bridges converges P for P almost every omega so P almost omega or convergence also in L2P norm okay so so Brunel and Kin have reminded the reason that Brunel and Kin had when you write it this way you get the Carti's function of AT and X and then your GSN of Y if you apply Y to this we are looking for the point-wise convergence with respect to Y of this single set of full measure where you take this little x clear here any questions so far no okay so the general case for T just the measure transformations S measure preserving transformations was done by Bourguin also in 1989-1990 and the proof that he had was a 50 pages long in fact at this time it's from AT several mathematicians were asking questions to Jean Bourguin and he sent regularly every month solution 50 pages long papers okay so this one was not an exception it was also 50 pages long paper but was never published at the end because by the time he wants to submit to to publish it he got another proof with a first and bear cut sense on Bourguin first and bear cut sense on a north side okay so now let's see for the return times and I'm giving all this because I'm giving the slide to presentation science to Stefano so if some of you are interested they can get to copy of this slide with all the data at the end of the slide there's also a list of references so you see exactly what was done and who was so that's when we get this number 24 one and all that so you have all the references there in case some of you are interested so I said that awesome crowd proof that you have so Rudolph did the multiple return times now the I was interested in already 1919-1995 after meeting first and then he brought all these questions to to my attention I was getting a result on a multiple recurrence point wise and I start mixing result in multiple recurrence and multi-term return times in 2000 okay so it was even before the awesome crowd the good factor and everything like that so awesome car show that is good enough to wait for the non convergence in 2009 this is exactly what they did here so instead of having here g of s n what they did is they put here in front of that one of the n the song from n equals 0 to capital n minus 1 the castle function 80 and x and then let's take the case g s 1 s g 1 and g 2 s to n and to be nice and have a fun number here that's right and they show that they exist a set of full measure where no you can extract from that a set capital X a where these averages are going to come there for basically no all dynamical system and all function g 1 g 2 g 3 which are bounded okay so this is a generalization of the result that they had already when here this weight was replaced by one so as you can see this fits with where this kind of universal weight that I mentioned at the beginning when you have a process the process in this case is this is a process x n and this is your weight which is c n here and here's your process and then you have regime now that's what you are doing okay so it's a non-convergence result right so if you take g 1 g 2 g 3 equal to the classic function of a set e that's what you get basically for the case no replacing 3 by k okay can you generalize that well if instead of returning to the set a along the powers of t you could try to have a weight of a form c a 1 t n x I'm just looking at the simplest case c a 2 t 2 n x that's the kind of wait c n that I'm interested in so let me get that so we know that this kind of weight the average is just so this weight converged by Morgan double-acryl theorem and in 2001 my PhD student Debbie Duncan provided what you call a winner in the result for that so not going to explain too much because I don't have too much time what the winner in the result is but there is a generalization of this result where you replace here you multiply this by you take the casting function of a t n x the casting function of a 1 a 2 t 2 n x and you put here a weight e to a 2 pi i n t so it also difficult to show that for every x for every t this is going to converge but the key is to find a set which is independent of t a full measure which is converged for every t now why is it interesting it's because if you have something like that then it's just a simple application of a spectral theorem will give you automatically the convergent in norm of these averages these two things are equivalent so if you have a commercial store of the single set for all t this is equivalent to the non conversions for this so the first step was to try to get a winner in the result for the double recurrence it was done by David Duncan in 2000 we've had no at the time the existence of the awesome craft factors so now in 2014 I'm going to start no quoting now the result that I have with Rio all of them are been referee all of them are accepted to papers no it is yes you know that is a mathematical that four papers on this subject for okay so is a point of working well because I keep okay yeah okay so 2014 this is a result that we had the first result was to as I said get this winner in the result this winner in the result which is to say which is generalizational Morgan double recurrence that many people were stunned to get because they thought that the bogey result is so difficult in term of his proof that they thought that it was a very difficult result to get actually is about 15 pages long to get this result it's not that complicated if you use a nice a nice approach so we get this one where the case okay Austin correct Austin class equal factor I don't have time to explain what the Austin class equal factor is but what you have here is that the result works the winner with the result works for the double recurrence so this is already an extension of Morgan result self which was not considered the trivial result okay they said this is already to show that no this kind of weight are good universal for the non conversions for any dynamical system now a little bit about this because of the fact that I'm sending the slide for those who wants to get a big quick glance at what was done so I'm not going to talk about an elm step in importance know the group but they are all at the basis and behind the scene in term of a proof there and what is the co-compact what is the else what is a new system I'll just cruise on that okay and that's the question on it no okay okay so now in 2014 after hearing this result here that we had this one that once converge for a great tea Benji vice ask okay can we replace no this weight e to the power NT replace this e to the power NT by just what is called a new sequence a new sequence is a more general form of this kind of weight and he asked whether or not we could replace e to the power NT by these new sequences okay so give you a definition of what the new system is a bit complicated but just for you to get an idea general idea what is there so this is what is called a one step new sequence and in fact if you report here instead of a point of a degree one which is NT you take a polynomial which is with integer values this is also e to the PNT is also an L step new sequence now the interest in your sequence come from number theory which is something that you see in the work of Green and Tao many times several papers on new sequences also here I'm not going to to get into the detail of that okay now I was able to answer the question by Benji vice positively and by showing that we could replace and a generalize this by putting here instead of e to the point E use here BN where BN is a new sequence so this work was obtained almost at the same time by Pavel in fact Zering Kranich Pavel Zering Kranich also obtained this result I think that we both posted the result on archive more or less within two days each other so okay so we have that then now we start getting greedy so instead of having a winter then let's try to have something a bit more general and putting here this instead of just decompose the SN can we just get no this type of process in front of that and if you get this process this means that you'll be at the same time a generalization of bourguer and a generalization of the awesome craft result we are able to get that too so let me just give you that that's right we've got also some data on what was done before be sure so 1990 you have the zing fancy kina case just for the polynomial case you have also eyes near and and cross cross is a student of Tao tiny eyes near you have also Pavel uh so we prove this result here I'm going to just click here again in 2015 which is at the same time contains bourguer result and also the awesome crowd result which is to show that this CN are a good weight for the first 10 bar pressures so that was one and let's be more greedy let's try to see if also if you take the commuting case you can still have the same result the answer is yes right so that's what I'm saying here so we can at the same time get the bourguer result entire result at once by the following I'm going just to say that here again 2015 well as you can see instead of having here the power of of s to the i n we get si to the end where si are commuting transformations so if you take f1 and f2 to be just a functional identically equal to 1 you get no but no the first 10 bar pressures but for commuting transformations in this case no we get a result which at the same time generalized bourguer and also the tower result which was considered for many as out of reach for at least a couple of years okay so now let's ask some questions no any questions there because this is quite technical I don't want to be both more boring but I need to I give you too many technical details just for you to get know the essence of the result and the references anyone interested in getting more details I don't think so we are close to lunchtime and I understand so let's close on that but quickly and enjoy the lunch okay so okay just give you an a couple of ideas how to do it so now you get the winner winner on one hand and the second one is to look factors the appropriate factors what are the appropriate factors of why that's a basic idea decompose that into two steps one step is to look at the winner of the second one is to look at the appropriate factors of why okay then no if you take your complement of a factor you get the non-convergence to zero if they all belong to the factor then you apply a Libman convergence repair so this is a simple path but there are some technical problems along the way because you have to have no set which are just independent of the other processes and that's what makes the things technically difficult okay so now next steps next step is to try to get a legal offshore here instead of having no transformation which are commuting here what we would do is to take no transformation we formulate within group which is the most general form of that that we don't have at yet so that's the next step that we are trying to look for okay now go back to the problem about understanding both the structure of set of arithmetic progressions is no looking at something like this is it strictly positive is the limit strictly positive and it exists of course you must have ergo this city in some ways to be sure that the limit is strictly positive for just to wait but this question is still open there's a limit is also strictly positive okay and um is there a lower bound is it synodicity so all this kind of stuff which are application to number theory so I have a result which is more general than those who great know this is the existence of set of positive upper density uh generating arithmetic progressions but so far we are looking at the applications to number theory clearly there are already something that you can understand about the structure of uh set of uh of subsequences but set subset of uh of natural numbers generating or containing a lot of arithmetic progressions but no this is a much sharper result okay okay here are the references no give you references there uh all of them are there uh submitted Rio Brunel Deli Duncan, First and Bear, Le Zing, Miguel Walsh, Pavel Zorin and I think before that also okay after that thanks for your family mutations okay