 But it is related in a sense because the conference is mainly concentrated on various random structures, but this will be about quasi-periodic quasi-periodic matrix, which is sort of a transition between deterministic things and random things, so I think it's sort of relevant in the sense Okay, so So and this will be the most well-known quasi-periodic operator because it's the simplest one and it models electron on a two-dimensional square lattice in perpendicular magnetic field So it's sort of lattice potential and the magnetic field on top of it and that creates possibility for quasi-periodic phenomenon Okay, so this is the we just look at this the following operator This is H as an operator on L2 of Z It's sort of given by the given by the following expression To lambda cosine Okay, so this is a okay. So as I said, it's models 2d electron So on the square lattice in perpendicular magnetic field so The parameter alpha Comes from the magnetic field and the parameter lambda is a hopin parameter on the lattice So if lambda is equal to 1 we have uniform lattice Square lattice and if it's not one then one direction is preferable to the another direction Okay, so this leads to some interesting Phenomenon this competition between magnetic field and the lattice So namely, we would like to ask the question. What is the spectrum of this operator? How it looks like and then we observe the following that for Irrational irrational alpha we have Following cases so first lambda larger than one then it turns out that This corresponds to the so-called localization which means that the spectrum is pure point The pure point spectrum with exponentially decreasing eigenfunctions And this is true for almost every alpha and theta But not for all So if lambda equals one we have a so-called critical critical regime Critical critical regime So here we have a singular continuous spectrum and this also as I said lambda equals one corresponds to the Uniform uniform lattice that is where this interactions in both directions are the same Sort of this is the most interesting therefore case. I think Okay, and lambda less than one so lambda between zero and one So this is a delocalization Delocalization It corresponds to absolutely continuous spectrum or all alpha and that So this is a kind of understandable in a sense that if lambda is very large then this part dominates And so it's obviously pure point and if lambda is zero it is sort of a cosine as eigenfunctions So it's a capitol of the continuous, but the transition is exactly at the point lambda equals one in this sense Okay, so but this is a So the next thing is that we will be so I will be kind of explain you what happens To the spectrum as a set So well, I just look at the spectrum of H So we will denote it simply Sigma of H So as a set this is bounded set on the real line and So how does it look like right? So so first of all if alpha is rational Then it is a standard standard result for Periodic operators that it consists of Q intervals or P over Q theta This Q intervals General they can touch but in this case they normally don't touch so if Alpha is a rational I will just so far remark that that this Sigma of H alpha theta is independent of the phase theta Yes, and in fact looking back here physically significant is not the spectrum Not this spectrum, but rather the union of all set us because the the the proper parameters alpha lambda and this This is a kind of Auxiliary so we would like it to be we're interested in the union of all over set up but for alpha irrational doesn't matter because It is as I said doesn't it does not depend sort of adjust their Godicity property of this Of the same, okay So we will also denote denote Billy back measure of this Set the following way so now we can formulate a theorem which says that for all Irrational alpha The real Theta and Lambda no not equal to zero the spectrum is Is a counter set of measure one minus lambda? One minus long four times one minus long Actually, we can restrict our attention to positive lambda because it's very simple transformation relating positive and negative Okay, so This Serum is a very very long history. It started somewhere in In the 60s already. I think in the 60s and so now this kind of it used to be known as conjecture So the measure conjecture used to be called Aubrey Aubrey and andre injecture And the counter set conjecture became known as a 10 martini problem because Ten martini was apparently offered by mark cuts for anyone could to solve this conjecture and and Barry Simon then Propagated this this name and so it became now it's sort of Called ten martini problem He's a problem and now I will give you some history of the of the proof of this Some history of the proof Okay, but first I would like to make a kind of a note for the critical case if lambda is equal to one So then the Lebesgue measure is zero and it follows from here immediately that The spectrum is a counter set. So because the counter set by definition is Closed no, but then said to that isolated points I Absence of isolated points for this kind of from the general or Gothic series Can be reduced and any spectrum is of course closed and the only thing which remains is this nowhere dense So basically the statement is that it is nowhere dense Okay, so and if it's measure zero then immediately it's clear that in nowhere dense. Otherwise, it's cannot be Okay, so but if lambda is not equal to one of then it is not clear because it's positive measure So it can be a counter set that may not And can happen Okay, so the history of the of the proofs Maybe one of the first rigorous result was to show that the sigma is Contour set for a dense set of pairs Lambda alpha was done by Billy Sardin Simon. I think it was 1982 okay, so then Then it was shown the next that sigma is a counter set For alphas which are very quickly approximable by rational Approximations so they they are they are called I call I call them sort of vaguely willion alpha So for you know alpha there they have a measure zero the typically this is which are proximal by rationals Faster than exponential exponentially quickly fast so So so this this was done By joy Elliott and and you in 1990 it was very kind of important development Okay, so then then there were some some works on the Continuity of the spectrum with the spectrum of irrational is approximated by spectrum of rational Approximates from here it came it came a very nice result that sigma That the that the measure of the of the sigma Get a measure of sigma is equal for minus minus lambda or almost every For almost every alpha and I can be even more precise and say which which alpha Actually such that that the limit when you approximate by by the rational Approximates your alpha So then then this limit should be zero So these are the irrational switch have coefficients of a continued fraction expansion from an unbounded sequence and They form a set of full measure and for that for that last I proved this formula in 1993-94 here's the two papers. So in particular if lambda is One If lambda equals one then it follows that there's zero measure Okay, but but then there was there was there was a problem in his proof because well Not in the proof, but he couldn't prove it for for all alpha because you know They didn't have enough Strong enough continuity for for the spectrum and so that remained for a while unsolved Then then actually that's what that's how I started with with this Well, it's not started, but but then we were now paper was Lana Zhitomirskaya We actually could show that this the measure is Given by this formula for lambda not equal to one like this and Dollar-Rational for the rational alpha. So this is our paper was in 2002 I think Done, okay, so Yes, so that that then we use the fact that In sort of supercritical regime we have exponentially decreasing Eigenfunctions that allowed us to improve continuity. Okay, but then it wasn't clear how to how to Do it for the case for the critical case first of all and then it wasn't clear How to answer the question about the counter structure and Here the the next step was to show that that sigma For lambda not equal one is indeed a counter set or for some diaphanthine Irrationals, so this is done by push in 19 in 2004. Yes So this diaphanthine irrationals are those which are poorly approximated by rational So as I'm with the golden mean is a standard Representation says standard irrational this time But it's so they form this this form set of measure zero Okay, so So then then and still it didn't didn't didn't cover the case lambda equals one so then then The fact that the measure of this sigma is equal to zero for lambda equals one and And all the rational alpha the rational alpha so was done in the work of Avala and Gregorium 2006 so here it has some Significance for the for the continuation of the talk. So sort of here they used they used The theory of dynamical systems in some sense so dynamical dynamical systems It is rather lone paper since 30 30 pages 40 pages So to to extend so to extend the result the result of last diaphanthine diaphanthine Alpha so the argument works there for for diaphanthine alpha and So together with the with this kind of result of last It settles the settles of the case of lambda equals one and so what remains what remains here is just to to show that it is a counter set for for the rest of Irrational alphas when it is not equal to one right so then it will be and this was completed that Sigma or Lambda not equal to one is Country set for all for all rational alpha so and this was a joint work of Avala and Gtomerska in 2009 I think Okay, so this is a this is a brief history of The proof of this of this property So now so we have now a new work with sweet Lana Where we found a much easier argument for the critical case So so that that I would like to present a little bit. So from now on we could see the only Cedar the case Lambda equals one Everything which I will be telling you from now on will be about lambda equals one and I remove your lamp Okay, so and in fact what we what we do that in this case we we actually give two elementary proofs of The serum for this case lambda equals one. So Namely, we show that Sigma absolute value of Sigma is equal to So measure of Sigma is zero for all irrational alpha. And so our argument work at the same time for all irrational alphas Rather than separating actually works so argument works and so the the idea of the of the proof is just to Use a special unit retransformation Which transforms this operating in a different form and this Kind of utilizes the fact that the problem is two-dimensional and you can the right the magnetic field in different gauges And this is sort of corresponds to the change of the gauge of magnetic field But we will write a couple of moments later on the the new representation Okay, but so what what new can we do this is kind of reproof of the known result, but we could Also prove something new and it's also an interesting Interesting question It has also long history From the 80s and so the question because it was conjectured for a long time that for this critical case the spectrum is Has a zero measure. So the natural question, of course is what is the what is this set? How can I characterize a set and the the most? Obvious question is what is the house dwarf dimension of the set, right? So so question question What is house dwarf dimension of the sigma? Okay, so So here I Maybe I will remind you about the well just the definition of the house dwarf dimension To make it clear So first we first we define the so-called house dwarf measure Responding to parameter t of a set a so a is a subset of the real line So and we define measure in the following way, so that's a limit on delta tends to zero of the infimum of the sums what value E so a is covered by this intervals and This is a delta cover Take a infimum of all delta covers and then you take the limit when delta tends to to zero And so what is a delta cover that it's just that here why these are the interval intervals of length Less so equal than delta, okay So if you look at this object, you will see that very often it will be there is a zero infinity and in fact If t is very small then it will be infinity And if t is close to one it may be maybe zero And so there there is there is a certain point where it switches from zero to infinity and this point is known as a house dwarf dimension so house dwarf dimension of the set Can be written as an infimum over t such that this house dwarf measure is is finite some point it will Start becoming infinite and this is it. So if we have a set on the real line this Dimension will be somewhere between zero zero and one zero and one right, so this is the the question So now So thawless 1980s So by by numerical numerical computations So observed and as he observed that actually measure of the spectrum on the in the rational case p over q and union over all factors This measures it behaves some constant divided by q when as Alpha is a as p over q approach itself some irrational So it is Rather small and this led him of course to the conjecture again that it is which was Slightly earlier that it is kind of zero that for alpha irrationalities as zero measure But of course, you know, there is a question of Proving that and a question of proving continuity and there are two questions so So if if we assume that if all intervals are of the same size If all q intervals Of the spectrum of this rational operator The were Of same size Which is in reality not not so But if they were all at the same size The size of one the size of one would be or is some constant over q q n squared or q squared Or q of them q squared and So this suggests That the household dimension is one half Because if you take it to the power one half and multiply by q the number of integrals, it will be a finite number And if you take something larger than one half, uh, then It will it will be small So currently zero and Smaller it will be infinity or it will be something it will be Something growing with q Right. So sort of intuitively it it would follow that So So one Injections that House dork dimension of this sigma is one half For irrational alpha So this is Unfortunately, it's not true In general, it may still be true in some sense Maybe for almost all but in general it is not true because there are exceptional cases. So, uh, so they exist Leovillian alpha Such that this house dork dimension Is actually zero It's very very thin set So this was, uh shown by by last And uh shamis Rather recently in 16 But on the other hand, uh, it is also known that they exist. They exist, uh some Daffentine alpha Such that this dimension is positive positive So this was uh done in the work of uh, elfer With uh collaborators I think you and Joe If I'm not I might miss a name here. Okay, so So in principle We have this a little bit mixed picture on the other hand. It was shown by last that that actually Or Some uh Daffentine alpha this dimension Uh is less Is less or equal than one half It was done by last in 1994 Okay, so The apparent there is also there also a conjecture apparently due to billy sar that actually Uh, it should be, you know, the same number for almost all irrational alphas Okay, but uh, what what did we prove there here is that we prove the following theorem So So we say that for all Irrational alphas A real theta So this dimension Is uh, actually less or equal than half For all irrationals So, uh, as I said, we do not expect it to be equal to one half It might be that it is equal to certain number less than one half or equal to one half for almost all that would be reasonable Okay, so So now, uh, maybe I will give you the main element of the proof Because there are several elements Give you the one which sort of gave us the right idea Uh, how to proceed Proof is the following We know that there exists a unitary transformation Which is uh, which is explicit That that shows the following that uh, the the union of the spectrum of h Two alpha theta is equal then the union of Of a set of spectrum of h with head alpha theta Where the operator with head acts the following way So it it's a gives you two sine two pi Alpha n minus one plus zeta Times v of n minus one plus two sine Pi alpha n plus zeta Times v of n plus one And that's it. There is no diagonal. There is no diagonal term and uh, so why uh, so this first of all physically This corresponds to the change of the gauge of magnetic field. So it sort of utilizes the two dimensional structure of the of the model And why is it uh, why is it good it is good because sometimes this, uh Objects become very small for some n This becomes very very small sometimes And here uh, here there is no chance because it's one and one here. So diagonal elements sometimes become very small And so if you look at this matrix Like this For example, this is small and this is symmetrically small And if this is very small you can see that it's practically splits into finite dimensional blocks So since uh Of diagonal elements can be small So this h hat of let's say quasi separates So which allows to obtain a better continuity statement for the spectrum So which allows to improve continuity result on the spectrum a result Uh on the spectrum on the spectrum as h p over q Sort of approximates approximates h alpha And from here we after that after that one can After this statement is improved sufficiently one can use the the argument used by last before in 1994 So and from here from here So the the result follows basically from here the fact that absolute value that Measure is zero and dimension sigma Less than equal one half follows So in fact, I should say that This is much easier statement the measure zero for that only a tiny improvement of continuity Was sufficient But in order to prove this much it's a much stronger improvement improvement of continuity Yes, so this is a I think uh should I finish already or I have Uh-huh. So maybe I just tell you what about about this continuity a little bit because it sounds a bit mysterious So Yes So for that I just tell you how last obtained his result So last On that for Rational case or the rational case The spectrum the union over theta Is between something bounded some constant over q some other constant over q is explicit constants Something like eight times e some pi times something in this constant So which combined was a combined which which is which which has combined them with continuity statement following that For every point in the spectrum of the alpha Irrational There exists a pointy prime in the spectrum of Rational approximate union over theta Such that the distance between e and d prime Is less than some constant times the square root alpha minus p over q That is a continuity statement, which was obtained by simon van moosh And avran previously So what it means is that So suppose you have the your points of the spectrum of your presumable Contour set of irrational Irrational situation irrational h alpha And so it says that they cannot be very far away from the from the intervals of the rational approximate So we have intervals of rational approximate and must be somewhere over here And and this and this boundary where they can deviate The size of this deviation is of order square root alpha minus p over q This is what it says simply And so then it gives a simple bound for the for the measure So sigma of h alpha then Is bounded by uh sigma of h p over q Over union over theta Absolute uh yes Over union over theta of this Plus uh the error. What is the error? The error is is this interval Times the number of edges The q interval to q edges. So so it's to q Some constant and the root alpha minus p over q And and that is it because This thing He showed us one over q. So it if you increase q it it becomes smaller and smaller And this thing will will tend to zero As p over q tends to alpha and q q increases for almost every Alpha that's why I wrote your condition before but not for all So in other words the right hand side of this inequality can be made arbitrary small From which is follows that the measure this one should be zero So it then follows that the measure is zero. So it follows that measure of sigma of alpha theta Actually zero For almost every Alpha so this is an argument of last for the household dimension one can Use a bit somewhat similar construction Okay, so if we if we can improve this Then then of course We will we will have that it will be true for for all for all alpha Because even a slight improvement of this root there is one half plus epsilon This is then true for all the rationals All the rationals and so but we we actually we show that Certain sense often We have this e minus c prime Where so equal some constant and the first power of alpha minus p over q slightly worse with some logarithmic Badness here. Yes. Not not for all points, but and not for all Thetas but often And so that of course immediately Sort of if you q is multiplied by absolute value that it is goes to zero for all Irrational and also it allows us to To have this continued to household dimension statement as well as continuity and this continuity We could only obtain because we had this nice structure of the matrix with matrix elements being very small Which means that uh, you can So technically here you need to cut off the test function and it is very nice to cut off Exactly at these points where you know, it was a very little error Because in a previous representation, uh, it wouldn't matter if it's always one Okay, so that's why so it works. Okay. Thank you very much