 Hello. So, in the last capsule we can completed the proof of the theorem of equidistribution of Hermann Weyl. As we saw it is a sharpening of the Kronecker's theorem. Now we will see certain ramifications of equidistribution modulo 1 of Hermann Weyl's theorem. So, we may have suspected that the result of Hermann Weyl has some connections with probability theory. In view of the fact, the Cesaro sums that you saw, the averages that you saw gives the expectation of the random variable f of x. And this together with the uniform distribution lurking in the background may have led you to believe that Hermann Weyl's theorem itself would be a special case of a more general phenomena. This is so and we shall demonstrate this through some examples. We cannot carry this to completion because that will be another course altogether. Here what you will give you is only a trailer of the full movie. So, first we define the notion of a normal number. So, let us look at numbers in the interval 0 to 1 and each real number in the interval 0 to 1 can be written in decimal form x equal to 0. a1, a2, a3, etc. where a1, a2, a3 are the digits of the number. Let us pick up some digit say 7 and let us ask how many times does a digit 7 appear in the string a1, a2, a3, etc. Let us look at the first n digits a1, a2, a3 up to an. In the first n digits, let kn be the number of times the digit 7 appears. And then let us take the ratio kn by n and let us ask for the limit kn by n as n tends to infinity. First of all will this limit exist and if so what is the value of the limit? This limit would be the asymptotic relative frequency with which the digit 7 appears. We say that the number is normal with respect to the decimal system if all digits not just 7, but all digits must appear with the same relative frequency same asymptotic relative frequency 1 by 10. So, this limit must exist and it must be equal to 1 by 10 and it must be 1 by 10 not only for 7, but for all the 10 digits. Then you say that the number x is normal. Now we state the theorem of Borel. The set of numbers in the interval 0, 1 that are not normal has Lebesgue measure 0. In other words, almost all numbers are normal. This is a celebrated theorem of Borel. Of course, there is nothing special about the decimal system. You could work with any radix. For example, you could work with binary systems and you can ask what is the asymptotic relative frequency with which the digits 0 and 1 appear and they are supposed to appear with asymptotic relative frequency half each. Then you will say that x is normal in the binary system. You can work with hexadecimal system, there is radix 16 if you like whatever. So, that is the first observation, but let us for simplicity work with a decimal system because that is the most familiar one and the arguments will go through and you will figure out how to modify the arguments appropriately for other radix. As an exercise, I want you to come up with a number x for which this limit does not exist. In other words, the limit itself does not exist. For example, suppose I take a rational number say 0.333333 infinitely many strings of threes is obviously not normal because the because the digit 3 appears with asymptotic relative frequency 1 and all of the digits appear with asymptotic relative frequency 0. So, rational numbers are unlikely to be normal except very special ones and I will leave it to you to figure out which rational numbers are going to be normal. So, please determine all rational numbers that are normal. So, let us look at the Borel's theorem in the light of what we are done. How is it very similar to Weyl's theorem on equidistribution model 1? So, to bring out the similarity, let us define a transformation from the closed interval 0 1 to the closed interval 0 1. What is this transformation I am talking about? Take the number x multiplied by 10 and take the fractional part. Multiplication by 10 simply shift the decimal point to the next and then I am knocking off the integer part and I am taking the fractional part. So, Tx of 0.a1a2a3 will be 0.a2a3a4 etc. Now, let us assume that this number x is irrational because I already told you that very few rational numbers are going to be normal and I am going to find out all of them. So, we may as well assume that x is irrational. Rational numbers anyway have Lebesgue measures 0 and so, the digit a2 will be 7 if and only if Tx lands up in the sub interval 7 by 10 8 by 10. Now, remember that if you take a number between 0 and 1, the first digit after the decimal point will be 7 if and only if that point lies in the interval 7 by 10 8 by 10. I just taken 7 for simplicity and for illustration purposes you can take any other digit if you like. So, a2 will be 7 if and only if Tx lies in this interval. Similarly, a3 will be 7 if and only if T squared x lies in the interval 7 by 10 8 by 10. So, we have a nice recipe for deciding whether the digit a4 will be 7 or not that will depend on whether t cube x lies in that interval. So, now it is very clear what we must do our kn by n the number of times the digit 7 appears in the first n strings is simply the arithmetic mean of the characteristic function evaluated at x with the characteristic function evaluated at tx plus zeta plus characteristic function evaluated t to the power n minus 1x where chi is the characteristic function of this interval 7 by 10 8 by 10. So, this open interval is the interval i and I have taken the characteristic function of that interval. We are asking the question whether this limit 1 upon n times chi of x plus zeta plus chi of t to the power n minus 1x whether that is going to be equal to 1 by 10, but 1 by 10 is a integral from 0 to 1 chi x dx. So, this is the equation that we are looking for and the question is whether this equation is valid for almost all x in the interval 0 to 1. Now, again the first thing to do would be to replace this characteristic function chi by integrable function f and ask the more general question whether limit as n tends to infinity 1 upon n f of x plus f of tx plus zeta plus f of t to the power n minus 1x whether that is equal to integral 0 to 1 f of x dx for almost all x. When you formulate this problem in this language, then the formulation shows that both Hermann Weyl's theorem and Borel's theorem are probably special cases of a more general phenomenon. This is indeed the case and the general phenomenon is encapsulated by what is called as a strong law of large numbers in probability theory and even beyond this we can go and there is a theorem called Burkov's individual ergodic theorem in dynamical systems. We are talking about measure theoretic dynamical system. The latter arose from the study of recurrence in differential equations quantifying a famous theorem of Poincare called the Poincare recurrence theorem. So, these are matters that are studied in measure theoretic dynamics and the Burkov's individual ergodic theorem arose from the study of large particle systems in statistical mechanics. What happens is that if you look at this particular equation, if you think of t as a transformation of space, the t was a transformation of some space and what we are done is that we have taken a characteristic function of a certain set A and the transformation takes a certain point and it looks at the orbit x t x t squared x t to the power n minus x dot dot dot that is the orbit of x. The successive hits if you like if you think of dart games. So, in the first n iterates in the first n trials, how many times do you hit the bullseye the set that you are chosen and you want to understand asymptotic relative frequency with which that happens and you have taken the characteristic function of the set and you computed this averages. So, in some sense this average that you see on the left hand side should be thought of as time averages and the average that you see on the right hand side should be thought of as space averages. So, this equality is an equality between time averages and space averages and for what transformations does this happen, those transformations are called ergodic transformations and that is exactly the way in which ergodicity was studied as phenomenon in statistical mechanics, whether time averages and space averages are equal or not. Now, you may want to consult the books of Patrick Billingsley. Patrick's Billingsley's book on measure and probability, it contains a proof of the Borel's theorem on normal numbers, a fairly elementary proof of Borel's theorem. Here elementary does not mean easy, it means he does not use a sophisticated machinery. On the other hand, Patrick Billingsley has written a beautiful book called ergodic theory in information. It is a very delightfully written book, there you will find these things discussed in the light of ergodic theory, time averages and space averages and ergodic transformations and so on. Of course, I must also mention the classic book of Hardy and Wright on number theory and where you will see a discussion of both Kronecker's theorem as well as Borel's theorem on normal numbers. With these general comments we must close this module on Fayer's theorem and we must move on to the next part of this course namely Fourier transforms. So, now the next few lectures will be on the second part of this course. So, we are leaving Fourier series for some time and we are passing on to Fourier transforms. Thank you very much.