 It's quite an honor to be asked to give a number theory seminar as I meant to imply in my abstract, I'm not really at all a number theorist. And in fact, the result I mentioned in the abstract is the, the only result I've actually published in number theory that can be in the sense that it can be mentioned they can be described without using any logic and really a mathematical logician and model theorist by training. So that paper as I said was published in the Journal of symbolic logic and the reason for that is that the main theorem there was this theorem I, I've written here so if you take any subset of the real plane, which is definable in no minimal structure. Now I'm sure many of you won't know necessarily what that is but you don't need to for the rest of the talk, but I'm just mentioning this as a piece of history. So if that's a subset of the real plane with empty algebraic part. Now the algebraic part of the set is just the union of all the connected algebraic curves defined over Q the rationales that are contained in X. So if there are no such curves. Then it's the case that if we just look at the integer points on the curve on X, which is essentially a union of finite union of curves. So the number of integer points of height age is bounded by h the epsilon for every epsilon with a constant there that depends on epsilon now, if you do know about open minimal structures you will know that we know much more than this now. I won't prove any more until I met a year or two later Jonathan Pila, and then we prove this same result for any definable subsets of r to the end and not just for integer points for rational points. So, it wasn't till I met Jonathan who proved similar results, functional points for analytics sets, and then we saw the combined things and got the full result that we wanted in 2006 I think that was published. Now but I'm not going to talk about that. I'm going to talk about the other result that was in this paper that I just noticed as an afterthought almost is that if your own minimal structure is our and you don't need really to know what that is it's just if X is a good what's called a globally sub analytic subset of our squared and I'll define that in a second, but it's a purely analytic notion. Then in fact the H to the epsilon can be brought right down to see log log H. So much much smaller the number of pairs integer pairs of points lying in a globally sub analytic sets with empty algebraic part is bounded by C log log H. So just to quickly recap, if necessary, what sub analytics sets are. Well first of all for a bounded subset of R to the end it's called semi analytic. If around every point, it can be described by qualities. Like it. What have I done here. Sorry, think what I've done here. Can you see. Oh, there we are. Okay. So let's look at the analytic if around every point. So this is for a subset of R to the end if around every point in R to the end. It can be described by qualities and inequalities between real analytic functions so if around every point in R to the end not just an S but in R to the end. So the neighborhood of that point such that as intersect the neighborhood is a bully a finite building combination of sets of this form those x is such that fi x is zero and g j x is greater than zero, where the f i's and g j is a real analytic functions on this neighborhood. So the sub analytic is the same, except the f i's and g j's now are allowed to be composed with inverse functions so they're allowed to be composed with y minus a divided by x minus a provided y minus a is strictly less than x minus a. I mean, that's not the usual definition but it's equivalent by a theorem of Daneff and Landon trees. So, well just if you haven't met this for before just to get a feeling of the difference between these two types of sets and there's a big difference as this example of Osgood from 1910 Osgood I think was a, an American mathematician from the 19th and early 20th century who did work on when a single variable differential, if you have a function of many variables in certain situations when. Differentiability with respect to each variable separately implies full differentiability I think that's one of it is one of his theorems but anyway he came up with this example. So if you take the triples of real numbers, satisfying these conditions so not less than or equal to y, less than or equal to x, and Z minus e to the y over x is zero show the graph of the function e to the y over x now that's a two obviously in some sense a two dimensional subset of zero three cubed, which is sub analytic by definition, but it's not semi-analytic on any neighborhood of the origin there's there's no analytic function of three variables which vanishes precisely on this set. So, and the e the exponential function there is that can be any transcendental function. Okay, that's semi and sub analytic sets, the global and that I just defined that for bounded sets but we're we're interested in integer points on these sets so obviously we need assets to be unbounded for a non trivial result. I would simply say that it's globally semi analytic or sub analytic. If the above definition applies to the image of the set under your favorite semi algebraic bijection between R to the end and zero one to the end so you just take any semi algebraic bijection with inverse semi algebraic and that and ask for that set the image which is now bounded to be semi algebraic semi analytic or sub analytic. And that's the definition of globally semi analytical sub analytic. Now, it's the reason I've mentioned this is that there's a proposition so this theory was first developed by voice a bitch. There's a proof that if you're just in the plane, then globally sub analytic and globally semi analytic are actually the same. And that's used in the proof of the of this theorem here this is why the proof works in that you only have to deal with semi analytic sets. And that theorem now just more or less amounts to the following, which I'll call theorem one. If you take any analytic function defined on the neighborhood of the origin in our squared real analytic function real valued real analytic function, who's zero said the outbreak part of the zero set is empty. Infinite semi algebraic curves in the zero set. And if H is a sufficiently large natural number. Then, by this inversion this row function you can take that to be almost anything. So here if we just take it to be the inverse function. And the set of integer pairs, such their inverse is lie in the domain of the function F, and the modulus of a and b are less than or equal to H. And if they happen to be zeros of this function F, then the number of such pairs is less than C log log H. Okay, so that that's really all the theorem I mentioned before amounts to on because. In fact, it's not quite as easy as that the F you have to take where you have to look at it zero set and that's described can be described but as a graph of a pressure series not necessarily an analytic function but that's a fairly small detail. And so the essence of the theorem amounts to this. Now, the theorems definitely false if you just take arbitrary rational that line you as an example of. Yeah, well there's certainly examples of an analytic function of one variable that takes a lot of rational values for rational arguments maybe each even H to the epsilon for epsilon going to zero. So the, the first result I mentioned is the best you can do here for rational arbitrary rational, but for inverse integers, you get this log log H. And presumably this is also false. If you take arbitrary sub analytic sets in higher dimensions because you can simply split up rational number to a pair of rational numbers as a four to pull up integers have to be a bit careful about the correct part but so this is where the matter rested for nearly 20 years 18 years or so I just couldn't improve this or find a decent generalization to other sub analytic sets. Okay, so but recently in discussion of the meeting with gal been the meaning and Gareth Jones. We reopen the case and we decided that rather than just dealing with arbitrary sub analytic sets or perhaps even arbitrary semi analytic sets perhaps it's this theorem I've marked with a star here is the one that you should try and generalize not be too ambitious. And just take an end variable analytic function neighborhood of the origin and just ask about the inverse integer points. I have to say we have no real applications of this result. Yeah, but anyway, we thought we'd just try and prove a generalization of this to end variables and just ask for inverse integer points close to the origin of zeros of analytic functions is there a natural bound on how many of those that can be. Now. So well first of all the word about the proof of this the proof of this uses auxiliary polynomials as in usual transcendence proofs. Where you convert information about a function, taking a large number of say a function taking a large number of rational values for rational arguments into another function that has a large number of zeros at those points. And then you can that polynomial you construct a polynomial involved with the functions and this is called an auxiliary polynomial. But then, Galbin you mainly observe that if you use the other way of constructing auxiliary polynomials which is to make another function you construct vanish to high degree at the origin. So using the information you have. So you use the different auxiliary polynomial methods. And observe that the following thing that if in theorem one if in this theorem. If you restrict your function F, if you restrict its Taylor coefficients at the origin to lie in some fixed number field, rather than just the arbitrary real numbers. Then you can actually reduce the C log log H to just see to a constant so he observed the following. In theorems once you dare to write it down it's actually not very difficult to prove. So if in theorem one, the Taylor coefficients of F at the origin line a fixed number field. And again the. This is still in two variables, and still the zero set of F still has zero algebraic empty algebraic part. And there's only a constant number of into inverse integers that can be zero. So for instance, I'm sure you look at that function. That function has rational Taylor, I mean the Taylor coefficients of that are all rational. There's only finitely many inverse integers that the zeroes of that function I'm sure you always wanted to know that anyway as I say. You get lots of trigonometric functions logarithmic functions so you can make but I, as I say we don't actually have a concrete transcendence theorem that follows from this, yeah, although we do have this constant. Anyway, so we decided to start using this new method of constructing not you, but you to us auxiliary polynomials around the origin to see what we could do and indeed we got the result we wanted so going back to a function, a real analytic function, defined on the neighborhood of the origin with whose zero set has empty algebraic part. Then for all sufficiently large age, we have the following result. If you just take the end two pools of integers of height less than equal to age so that's just the maximum modulus of the coordinates of a whose inverse so that's just the two ball consisting of the inverse of the coordinates. Okay, such that the inverse integers of coordinates are a zero of F that set of points has cardinality less than the constant, depending on the function time. That's meant to be times times there. So log H to the end. Okay. And we actually suspect you can do with then over to the integer part of it over to but that's not completely worked out. Anyway, so that's the new theorem, I wanted to announce. So I mentioned, well not this generalization but the original log log theorem. David Massaros, he invited me to Basel. And it was around the time he hadn't quite come out yet I think his book on auxiliary polynomials, and I told him I said well I have a result using auxiliary polynomials. But it's published in the Journal of Symbolic Logic, you might not have seen it. Yes, I've seen it. He did. He seen it and he said it's in my book. I said, oh, oh, that's great. Not being a number theorist I'm quite flattered by this. He said yes it's, it's, I call it your log log theorem. So not only do I have a theorem in David's book I have something called the log log theorem, which is even more status. So he looked he said yes he said yes your theorems in my book is in the exercises. So I thought, oh, and then he saw I was a bit disappointed so he said but it does have an asterisk. So, and if you look at some of the theorems that are in the exercises of David's book you'll see, I was probably in quite good company and shouldn't have been too disappointed by that anyway. I assume David's there. But anyway, I don't know if he saw that story like that. But anyway, thanks. Thanks for the plug. Okay. No need to put your hand up for that. Yeah. Okay. Right. So, so that's the theorem I call theorem one and we also have an algebraic number field version, which I call theorem to N. So this is the same hypothesis so we have a function defined in the neighborhood of the origin in real and space. Who zero set has empty algebraic part and we assume further that the Taylor coefficients of F around zero line a fixed number field. There's also something observed by galabini amini in his proof is that you, you don't need any assumption about the heights of these Taylor coefficients or even their modules well except. I mean there's a natural bound coming from convergence but there's there's no need to have any extra bound on the height as elements of the number field or on the modulus. Okay, given that what you can show is that if you just look at inverse integer points which are zeros of F and providing the coordinates don't differ too much they're multiplicatively comparable. Okay, so so the maximum of the coordinates of the modulus of coordinates is at most the coordinates to the power R. So if you take any say fix large R, then the number of such a is bounded by a constant where the constant has to depend on R. Okay, so obviously depends on the function and on our. So we have the many variable version of the theorem about how to take functions with with Taylor with Taylor coefficients in the number field. So that's true in n variables, providing you restrict the coordinates to not drift too far apart. As the as the points go to infinity or well they can't go to infinity if they don't drift far apart. I should say that in when n is to the version I proved above or mentioned above that naturally holds that is a thing called a voice a bitch in the quality that if n is to this is forced well for elements in the zero set. They can't drift too far apart even even the real numbers real zeros can't drift too far apart. Okay. So. So we actually get a finance theorem here. And it turns out that there's a better theorem than theorem one and namely the zeros here. The integer points are fairly uniformly distributed consistent with this upper bound so. We actually get a finite this theorem. So I call this theorem one and plus here. Because it's a it's a stronger theorem. There's no finite this theorem there's no log logs or no estimates or anything in this theorem. So it's the same hypothesis about that. And then we have the following for all are there is a constant just depending on our search that for all sufficiently large natural numbers h one up to a chain. So if you just look at the integer coordinate the integers, whose inverse is a zeros of the function, and which lie in a box like this. So, well let's just take our in equals to so the jth coordinate integer coordinate lies between hj squared and hj to the half. So suppose we just look at boxes like this, then the number of such points is bounded by a constant. And if you do take our equals to that clearly implies the log log theorem because you can cover. And then said to the end or points in art at the end even of modulus lesson or equal to age by about log log h to the end square cubes like this cubes of size. Now, more with multiplicatively dependent sides because. Well if you just take n equals one you a squared h squared h to the half and you go down to h to the quarter h to the eighth and of course you just get log log intervals. So, and you just raise that to the power n, and you can cover all of our to the end points in our to the end of modulus lesson or equal to our by just log log n cubes like this so you get the log blog result, but this is better because it says they are sort of more or less evenly distributed as as well. Oh, by the way, this is, I should say this is best possible or the even for n equals two. Well when you formulate it like this, this is best possible. We construct analytic functions of two variables in neighborhood of the origin, where each such square with our equals to say does contain a point that's a zero of f so the log log h is the best possible result. So the rest of the talk, I'm just going to try and sketch the proof and I'll do the number field case because it's a bit easier. And I'm going to do it in a slightly obscure way and being a model theorist I'm going to do it model theoretically I thought, although for the number field case, you don't actually gain a lot doing this way but what I'm going to point out is that it's much easier to we're just going to prove this a finiteness theorem and not one with a subtle log log h bound where we have to do lots of estimates. And so if you want to prove something's finite. Well, you can do it qualitatively I'm not going to try and do anything effectively here. I'm just going to prove that you get a contradiction from assuming something's infinite. So, in other words, suppose we have a number field like that and we have a function. Okay, they're just the power series or the convergent power series whose Taylor series have points in K and non not identically zero and for non triviality f of zero zero. And suppose I'm not assuming the condition about the algebraic part for this. Okay, that will come out in the in the wash. So suppose we have a sequence of integers of integer n tuples a one j up to a n j. And suppose that there's some function h of j mapping natural numbers, natural numbers to natural numbers which tends to infinity. Such that each of these coordinates lies between h of j and h of j to the R so it's, I mean you could take the minimum of the modulus of the AI j's. Okay, so suppose we have this fixed function going to infinity such that all these coordinates lie in the kind of a multiplicative class of the function age. Okay, and suppose further that suppose further that all the inverses of these integers are zeros of f. Okay, now, in general that can happen but we want to show that it can't, it can only happen if f has a non trivial algebraic part. Oh, now it's quite important when we go if well I won't have time but to go on to the where f just has arbitrary real complex coefficients that we prove something stronger. So we need to prove something stronger to be able to do a good induction on then we need to weaken this zero here. And the assumption to it just being o of h the minus hj where a little h is a function going to infinity as j goes to infinity any function is j goes to infinity. We want to show, no matter how slowly, we want to show that if effort these points has this order of magnitude doesn't have to be zero. And we can deduce something we want, which I'll state in a minute. So this is a bit like generalizing the original point counting theorem of myself and Jonathan Pila. This is the way that Philip Philip Habaga did to approximate point counting. This is roughly the order of magnitude that he shows the point counting theorem holds for you don't need your points. But he's dealing with arbitrary rational points but you don't need them actually to be on the zero set, but they can be this close to it and you still get your required contradiction. So what is the thing I'm going to be proving and this will take the rest of the talk, I shall prove. So there's no assumption here on the algebraic part of the zero so I shall prove that just under these hypotheses, the functions a one to the minus one a n to the minus one this functions from natural numbers to rationales are algebraically dependent over q. So, in other words there is a polynomial with rational coefficients, such that these functions are a zero of that polynomial for all J so a one minus what to the minus one of J up to a n to the minus one of J. So all J satisfy this polynomial. Okay, so the functions themselves, algebraically dependent, and that's just on these hypotheses that say they're zeros or that this is very small. So that's all you have to do to prove them to end because you do an induction on them. And you just reduce your set F where you have to do it with, I guess with algebraic sets not just one function, but you just add the polynomial to your system and it reduces dimension and it, it can't keep reducing the dimension. So if you go to zero and then you'll get an on trivial algebraic part of the original set. That's just like in the ordinary point counting theorem. So that proof part of the proof is well understood. So the crucial point is to show that if functions like this inverse integer valued functions, going to zero, tending to zero. They are analytically dependent in this sense. Then they're algebraically dependent. Okay, and of course you have to use some transcendence theory you have to use the fact these are in this integers, obviously not true for arbitrary rate sequences of rules. Okay, so the proof of this. So I should be able to do this in time. I'm going to do a fairly obscure proof now it's not too hard in this for this particular case the algebraic number field case. To do it directly using estimates and taking subsequences and do as various things but I'm actually going to do a bit of model theory here just because it's what I do, I suppose. Okay, so I'm going to suppose there's no such algebraic dependence. Okay, and so for each polynomial in the variable Z one up to Z then I let Z of P be those J such that this to pull a one J to the minus one up to a and J to the minus one is a zero of the poll. So there might be no such J or they certainly certainly can't be all such J because I'm supposing these are independent. Okay. Is that clear. That's just those J for which this to pull is a zero of P. And then I let I be just those for those subsets of natural numbers which are subsets of some ZP. Okay, so it's the X in n such that for some P, not identically zero X is a subset of ZP so I'm just closing the set of ZP's undertaking subsets. Then, in terms of Boolean algebra Boolean algebra nomenclature, I is what's called a front non principle ideal of sets. So in other words it satisfies these four axioms here. Firstly, if Y is a member of I and X is a subset of Y then X is a member of I that's that's by construction, that's obvious. I contains every finite set because any finite set of inverse integers can be made to zero of some polynomial obviously with rational coefficients. Now, crucially, if X or Maya and I then so is the union. Because if X is a subset of ZP and my is a subset of ZQ and X union my is a subset of ZP times Q. Okay. And if we assume this lemma is false that there's no algebraic dependence, then the natural the set of all natural numbers cannot be a member of I. Right, because that would give us the dependence we're looking for. But here is the non principle part of the definition contains no far it contains every finite set. Okay, so now we just take a maximal extension of I with these properties so that I till there'll be a maximal using the Zorn's lemma on the trade and that's crucial you can't do this without some form of the axis of choice, which might be distasteful to some but I'll remark on that in a minute. So let I tilled be maximal such that I is contained in I tilled so we just extend it to a maximal collection of sets satisfying one to four. And then it's a tie a very easy exercise to show that every set is either in it or his complement is. Okay, that's that's really just an exercise. And these sets that are in it I'll be calling small. And the sets that are not in it I'll be calling large. Okay. Now, if you go and look at the model theory literature. The set, the set of large that's here is sometimes called no filter on and that's the, if you want to look this stuff up. And this is called the maximum ideal of sets. Now, we're interested in properties of sequences and the trick here is to regard the sequence as an element of another field, the sequence itself, and then just do some standard field theory ring theory in this new field. Regarding the sequences is just elements from sequences. I mean this is this only works if you're just trying to prove a non-finimus theorem or a finance there and you, you won't get estimates like this, at least not naively. And so what we do to construct this field is going to be an extension of the kind of big algebraic extension, a big algebraically closed extension of the complex numbers. So when we first do this, I suppose F and G are any functions with domain and natural numbers. I don't care what that range is. We say F is equivalent to G, if you like here. If the J, such that f j quasi G J is a large set in the set that it's complement is in this idea. If I set S I look at all the sequences of elements of S. So that's just all the functions from natural numbers to S, and I take their equivalence class it always trivial this is an equivalent relation. Okay. So any function say from S one cross SP to S. I define its extension by till the to be the function from S one till the cross SP till the to S till the. I just define it in the only obvious way you can. I define F tilde of F one tilde up to F N tilde where this is in there and that's in there. So the function F of F one J if you like up to F N J as a function of J and taking that's that gives me a sequence J maps to F of F one J up to F N J and taking its tilde regarding it as a sequence of elements of S here. Am I doing the time. Then the theorem about this construction is called washes theorem, or possibly the fundamental theorem of non standard analysis about which I'll make a few comments. This star operator preserves all first order logical structure, such a property a logical property size true of the certain sets S one tilde up to SM tilde. Tilda world if you like if and only if it's true of S one J up to SM J for a large set of J. Okay, so it's. So let me just give you an example for example if L is a field then L tilde is also a field. So this follows from washes theorem because the axioms for a field of first order logical axioms, but you can see that this in the more algebraic way. And if I just look at all the functions, the sequences from now the functions from N to L such that their support is small. Okay, the points at which they are non zero is a small set in the above stands, then it's easy to prove that's a non principle maximal ideal in the usual algebraic sense of the ring. Let's go back into the L this is a ring with coordinate wise operations. And this is a maximal ideal in the usual sense. And you factor this ring of sequences from L by this maximal ideal you get this L tilde defined like above there. So it's a field because it's a ring factor by maximum idea. So by the way, we won't really need what first order logic is every time we'll use washes theorem you can just verify it directly, but the point is that if you just look at the usual inclusions of the natural numbers in the integers in the reals in the complexes, that holds true when you tilled or everything. Further are tilled or is a real closed ordered field C tilde is its algebraic closure just like down here. They're much bigger because we can embed the lower levels into the upper levels just via constant functions and taking that till that so C tilde turns out to be a very large algebraic closed field extending complex numbers. And in particular it contains the equivalence classes of all sequences into C. Perhaps a quick word, do I have time for nonstand analysis is usually, or in its infancy when Robinson Abraham Robinson invented it. It was thought to be a, or it is a way of rigorously introducing infinitesimals into the calculus so you rigorize liveness is approach to the calculus. You see this C tilde will contain infinitesimal, so will contains elements less than epsilon for all actual positive real numbers epsilon. One way to describe an infinitesimal is if it's S tilde. It just meant amounts to saying that for every ordinary positive epsilon there's a large set X. It's actually that SJ, the jth coordinate of S is less than epsilon for all J and X. But it's not quite the same as saying SJ tends to zero on a large set. It's more general. Anyway, so then you can the infinitesimal and there are even books written developing calculus using this nonstand analysis and this construction but of course, whether or not you I mean it's certainly easier if you if nonstand analysis comes to second nature but you probably have to explain that too. So I'm not sure about that but my view about it is that it gives us structures that you might not have seen before, and that can be useful when combined with ordinary completely standard mathematics. Anyway, let's forget that just for a second and get on to the proof of the theorem. So we have our function we're back in the standard world now on this neighborhood of zero. And I'm going to for convenience of the talk suppose that the derivative of F in the last variable at zero does not vanish. That's not particularly strong assumption. After a linear change of variables you can suppose that some derivative in the last variable at zero does not vanish. And then by the vice-trails preparation theorem you can actually suppose F is a polynomial, a monic polynomial in the last variable. And that's almost as good as this. Here we can of course use the implicit function theorem so we just take an implicit function phi that satisfies this just by the implicit function theorem and this. Oh, and by the way the five will have coefficients in K in the number field that's just by repeated differentiation of this relation here. So, in general this will be an algebraic function of other analytic functions but that's not too difficult to deal with. So we'll have this for all sufficiently large J here. Okay. So now we just construct our auxiliary polynomial. So this is all as I say in the standard world we just take a polynomial general generic polynomial instead one after ZN of degree D, or rather degree D in each variable. So this is where D is a large natural number. So we've started with the dimension of the number field K over Q as a Q vector space, and it's also got to be large compared with this number R, which compares the coordinates of the HA's, AI's. And the alphas are going to be integers not all zero. So we choose them. Well we first substitute phi of Z1 up to ZN minus one quite naturally right for the last variable call that function p star of Z1 up to ZN minus one. And now we choose the alphas. The function p star vanishes to a very high order so to this order here in each of the variables. And we can do that because so how many, how many derivatives do we have to make vanish it well it's this to the power n minus one. So that is D to the less than D to the end to D to the end right D to the end sorry D to the end minus one. That's less than D to the end and we have that available because there are D to the end values we can a alphas. And then as you do this just by linear algebra we have more equate more unknowns than equations, there's no need to use seagulls lemma or anything like that we don't care how big the alphas are. You just, you just choose them to make all these this function vanish at zero to this high order. And then we choose Schwarz's lemma, which tells us that for small positive T. If we take the complex numbers said one up to ZN minus one will only be interested in realness but we take complex numbers in the disk centered zero radius T. Because of this order of vanishing here will be less than or equal to a constant. Now the constant will essentially be the sum of the modular of the alphas. But we don't care how big that is times T to the, well the degree of vanishing so that's this number here so. So we can make that as small as that. So in particular, we look in the structure C tilde in the field C tilde and look at these elements these are now elements of the fields you see T order. And they're obviously less than this T because they're infinitesimal. We certainly have the modulus of this is less than or equal to the h tilde which is a bound remember the bound for the a n till those to the same exponent or negative. So, all these are less than or equal all these are less than or equal to one over h tilde. That was right at the beginning in our hypotheses. So this is less than or equal to this, just by the Schwarz lemma, essentially. Now I've written a tilde over less than or equal to that's because we're interpreting the ordering on real numbers in our tilde. So this is just what you get when you lift the ordering on the rails up to our tilde. The first thing is a rational number it were in the sense of q tilde. Okay, this is an element of our extension q tilde. And it's denominator since these are inverse integers is of most this. Okay, because the coefficients of the p tilde's are all integers. And, well, it's a rational number with denominators this, which is less than or equal to h tilde. There are n of them there's a D here, and they're all bounded by h to the R each one of these so their product is bounded by that. But this is much less, well, than the inverse of this, if you take the minus away. I mean, so since if this is non zero it's numerator is at least one that cannot happen because that would conflict with these two inequalities here so this is the usual auxiliary polynomial argument. So this forces this to be zero. So actually by the wash theorem. This happens in our non standard universe which means that the set of j's for which it happens in the standard universe is large. Okay, that's precisely the wash theorem. So this is just the set ZP. So, ZP is large but that's our contradiction because we actually constructed our ideal of small sets contains all the ZPs. Okay, so ZP is small so that's the contradiction that proves the key lemma let's go back to it. Got time. So whenever you have an analytic dependence between inverse integers between very small inverse integers that is one over a large integer, then you must have an algebraic dependence as well. It's not at all difficult to recant redo this proof without mentioning any non standard notions or ultra filters at all, but it's not so easy to, well, there are a few details here that I admit it like using the vice trials preparation theorem, which you use in the extended universe where it becomes very easy and various other little things but it's probably no part to do this proof without non standard notions. However, the log log for the real coefficients, the analytic functions with real coefficients. In my view, my co authors disagree, I think, but in my view, it's much easier to use the non standard formulation, because again you're proving a finite theorem, not not anything with subtle bounds in it. And there, the induction involves not just going to see till the, but then you have to do it again, and do see till the till the, and that will eventually get you your result like this, even when you have real coefficients. In this case, I have to say you do need the two sequel theorem for finding integers, which make linear forms very small, real linear forms you have to use that result but it's much I in my view easier using this till the framework. I wonder how many people listening would agree, but I'll stop there. Thank you very much.