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Uniform distribution of sequences: Transcendental number and U.D. mod 1

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Uniform distribution of sequences: Transcendental number and U.D. mod 1

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Content UNIFORM DISTRIBUTION OF SEQUENCES: TRANSCENDENTAL NUMBER AND U.D. MOD 1 by Yuxuan Gu A Thesis Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree Master of Science (APPLIED MATHEMATICS) August 2018 Copyright 2018 Yuxuan Gu Dedication This thesis is dedicated to my mom, dad. ii Acknowledgments With all the help received from the USC, I would like to convey my sincere gratitude to my advisor Professor Robert Sacker, for his guidance, advice, and support during my research. I am also grateful to Professor Richard Arratia and Professor Sergey V. Lotosky, for their valuable input as members of my examining committee. iii Contents Dedication ii Acknowledgments iii List of Figures v Abstract vi 1 Introduction 1 1.1 Uniform Distribution of Sequence modulo 1 . . . . . . . . . . . . . 4 1.2 Weyl’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Example: x n =λn(exercise(2.5) from [2]) . . . . . . . . . . . . . . . 11 1.4 Euler Summation Formula . . . . . . . . . . . . . . . . . . . . . . . 14 2 An analysis of sequence x n =e n 15 2.1 Weyl-Koksma’s Metric Theorem . . . . . . . . . . . . . . . . . . . . 15 2.2 Distribution of sequence e n mod 1 . . . . . . . . . . . . . . . . . . . 15 2.3 Quantile-Quantile plot for e n mod 1 . . . . . . . . . . . . . . . . . . 16 2.4 Combination of Euler summation formula and sequence e n . . . . . 18 2.5 Another approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Conlusion 23 Bibliography 26 A Appendix 1 28 B Appendix 2 30 iv List of Figures 1.1 gamma distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 qq-plot for distribution of en mod 1 . . . . . . . . . . . . . . . . . . 12 1.4 qq-plot for distribution of 3 2 n mod 1 . . . . . . . . . . . . . . . . . 13 2.1 distribution of e n mod 1 . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 qq-plot for distribution of e n mod 1 . . . . . . . . . . . . . . . . . . 17 2.3 absolute value of Ei(2πi`e x ) . . . . . . . . . . . . . . . . . . . . . . 21 3.1 qq-plot for sin(n)e n mod 1 . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 qq-plot for ( 3 2 ) n mod 1 . . . . . . . . . . . . . . . . . . . . . . . . . 25 B.1 code for the distribution and qq-plot . . . . . . . . . . . . . . . . . 30 v Abstract Ever since the Weyl criterion was introduced, many mathematicians tried to answer the question: Is there a transcendental number a > 1 such that a n is uniform distribution (mod 1) or not. In 1964, Woody Dudley proved that though cosnθ is not uniformly distributed (mod 1) for almost all θ, f(n)cosnθ is, where f is any function such that f(n) goes to infinity with n, no matter how slowly. In this thesis, the Weyl criterion is used to figure out a transcendental number a > 1 such that a n is uniform distributed (mod 1) or not. To help find such a transcendental number, a method called Euler summation will be introduced as a key to this problem. The transcendental number e will be used as an example in this thesis. vi Chapter 1 Introduction Mathematicians do not always study things because they are applicable; pure mathematics, often by definition, has no practical application. However, transcen- dental numbers such as π and e occur literally all the time in applied mathemat- ics; they are almost as ubiquitous as multiplication. They are clearly vital that most scientific problems cannot be done without them. According to NASA’s Jet Propulsion Laboratory [1], to calculate how much hydrogen might be available for chemical processes, and possibly biology, in the ocean beneath the surface of Jupiter’s moon Europa in a given unit area, we divide by Europa’s surface area, which is the area of a sphere with a radius of 970 miles (1561 kilometers). This is the use of π in the field of astronomy. On the other hand, e is widely used in the field of statistics. For instance, e is used in the formula for the gamma function which is a way to generalize factorial (see figure 1.1). Moreover, the normal distri- bution, commonly known as the bell curve occurs throughout statistics, includes not only e but also π (see figure 1.2). In mathematics, a transcendental number is either a real or complex number that is not a root of a nonzero polynomial equation with integer or rational co- efficients. The most well-known transcendental numbers are π and e. All real transcendental numbers are irrational. This means when applying a numerical method for transcendental numbers, accuracy, stability and steps of computations are the primary concerns for the users. Meanwhile, uniform distribution of se- quences modulo 1 (abbreviated u.d mod 1) is a numerical method distinguished 1 by the order of accuracy, i.e the study of convergence rate and efficiency. Accord- ing to L. Kuipers [2], the development of u.d mod 1 began with the celebrated paper from Hermann Weyl’s, the famous German mathematician, at 1916. During the last century, the theory has been a basis for topics such as number theory, probability theory, functional analysis, and topological algebra, etc. By studying the distribution of sequences such as e n and π n , we would get a wide view of the stability of each transcendental number. In the field of applied mathematics, we need somehow to convert our proof into applications. As a principle, since u.d mod 1 is a limit of counting processes, it basically helps to simplify some difficult counting problems in the mathematical field. For instance, counting homogeneous spaces and algebraic varieties of the distribution of rational points on algebraic groups needs the concept of uniform distribution of sequences. Algebraic varieties over a field n can have many points in an algebraic closure, but what if we want to know about the existence of points in a finite extensions? See Yuri Tchinkel’s method [3]. Another application with u.d mod 1 could be in physics. Quantum physicists pay much attention to the distribution of eigenfunction of Hamiltonians on mani- foldsastheprimalexampleofquantumchaos. Thephysicaltermiscalledquantum ergodicity. The definition of quantum ergodicity is basically a uniform distribu- tion statement. More importantly, Herman Weyl’s law on counting eigenvalues is involved in this subject. For this part, see Steve Zelditch’s method on quantum ergodicity [4]. Finally, this theory can be applied in the theoretical computer science. If you are a computer scientist and would like to use such applications, a book called the discrepancy method [5], written by Bernard Chazelle, a computer science professor 2 at Princeton University, would be appropriate. Discrepancy is a topic highly re- lated to uniform distribution of sequences. The topic selection of this book makes an excellent introduction to discrepancy theory in computer science and related fields. This thesis is composed of three chapters. Chapter 1 deals with introduction of u.d mod 1, Weyl criterion, and Euler summation formula. Chapter 2 provides an analysis of the distribution of e n . Chapter 3 provides conclusion and future study of this topic. Figure 1.1: gamma distribution 3 Figure 1.2: normal distribution 1.1 Uniform Distribution of Sequence modulo 1 Before talking about u.d mod 1, we should first introduce some definitions that helps understand notations of u.d mod 1. Let [x] denote the integral part of a real number x, that is the greatest integer less than or equal to x; let{x} = x−[x] be the fractional part of x. It is also known as the residue of x modulo 1. Note that the fractional part of any real number is contained in the interval [0,1). This concept shall be used in the u.d mod 1. Next, let (x n ), n = 1,2,..., be a given sequence of real numbers. For a positive integer N and a subset E of i, we define the counting function A(E;N;x n ) as the number of termsx n , 1≤n≤N, for which{x n }∈ E. Then let us provide the basic definition of uniform distribution modulo 1. DEFINITION : The sequence x n , n = 1,2,..., of real numbers is said to be u.d mod 1 if for every pair a, b of real numbers with 0≤a<b≤ 1, we have lim N→∞ A([a,b);N;x n ) N =b−a (1.1) 4 Remarks: 1. We can replace [a,b) by [a,b], (a,b], or [a,b] without changing the definition. 2. The definition is saying that the frequency with which the sequence{x n } (which we defined early this section) in [a, b) converges to b−a that is the length of the interval. 3. For better understanding of the definition, for example, if a sequence x n is uniformly distributed mod 1 in [0,1], since the interval [0.2,0.6] takes 2/5 of the length of [0,1], as n becomes enough large, we may assume that the first n elements of the sequence x n that fall in the interval [0.2,0.6] should be approximately 2/5. For a wide view of the whole sequence, we would like to say that each element of the sequence, which is uniformly distributed mod 1, is equally likely to fall anywhere in its range of interval. Next, we would like to provide a necessary and sufficient condition for x n to be uniformly distributed mod 1. This condition is called Weyl’s Criterion named after the German mathematician Herman Weyl. 5 1.2 Weyl’s Criterion Theorem The following statements are equivalent: (i) the sequence x n is uniformly distributed mod 1; (ii) for any complex-valued Riemann integrable function f on IR, that is f: [0, 1] → IR, we have lim N→∞ N P n=1 f({x n }) N = Z 1 0 f(x)dx (1.2) (iii) for each `∈Z\{0}, we have lim N→∞ N P n=1 e 2πi`{xn} N = 0 (1.3) Remarks: 1. According to Kuipers and Niederreiter’s book [2], (ii) is introduced as a corollary rather than a part of Weyl’s criterion. Condition (ii) is included as a close resemblance here. 2. For notational simplicity and without loss of generality, we may assume that x j ={x n } in the following part of the paper. Proof: (a) We first show that (i)⇒ (ii). 6 Suppose that x j is uniformly distributed mod 1. x j is an appropriate Riemann integrable function, see[13]. If χ [a,b] is the characteristic function 1 of the interval [a,b], then we write (1.2) as the following: lim N→∞ N P j=1 χ [a,b] (x j ) N = Z 1 0 χ [a,b] (x)dx (1.4) Next, we shall deduce that lim N→∞ N P j=1 g(x j ) N = Z 1 0 g(x)dx (1.5) where g is a finite linear combination of characteristic functions of intervals, i.e., g(x) = n P i=0 α i χ [a,b] (x) for all real numbers x, where n≥ 0, α i are real numbers. In mathematics, g(x), a finite linear combination of characteristic functions of inter- vals, is called step function. Now given an ε > 0, we can find a step function g with the condition that ||f−g|| ∞ ≤ε. We can calculate that N P j=1 f(x j ) N − Z 1 0 f(x)dx ≤ N P j=1 f(x j )−g(x j ) N + N P j=1 g(x j ) N − Z 1 0 g(x)dx + Z 1 0 g(x)dx− Z 1 0 g(x)dx ≤ε+ N P j=1 g(x j ) N − Z 1 0 g(x)dx +ε 1 χ [a,b] (x) = 1 if x∈ [a,b], χ [a,b] (x) = 0 if x / ∈ [a,b] 7 = 2ε+ N P j=1 g(x j ) N − Z 1 0 g(x)dx Since the second term converges to 0 as n→∞, we get that lim N→∞ N P j=1 f(x j ) N − Z 1 0 f(x)dx ≤ 2ε as we desired. Since ε is any real number greater than 0, Thus, we get that lim N→∞ N P n=1 f(x n ) N = Z 1 0 f(x)dx (1.6) which is what we want to prove. (b) By settingf(x) =e 2πi`{xn} , for each`∈Z\{0}, we directly get that (ii)⇒ (iii). Also, by calculatingf(0) andf(1), we get thatf(0) =e 0 = 1 andf(1) =e 2πi` = 1 (by setting h = 1), we get that f(0) =f(1) which satisfies the condition of (ii). (c) Now we want to show that (iii)⇒ (i). Suppose (iii) holds, that is, lim N→∞ N P n=1 g(x n ) N = Z 1 0 g(x)dx Here we define g(x) as a trigonometric polynomial (a finite linear combination of exponential functions), i.e. g(x) = N P n=1 a k e 2πi`x (a k can be any real numbers). Next, we let f be any continuous function on [0,1] with the condition that f(0) =f(1). 8 Similar to part (a) of the proof, given an ε > 0, we can find a trigonometric polynomial g such that||f−g|| ∞ ≤ε. From part (a) of the proof, we have that lim N→∞ N P n=1 f(x n ) N = Z 1 0 f(x)dx Then consider that[a,b]⊂ [0,1). Given anε> 0, we can find continuous functions y,z, with y(0) =y(1) and z(0) =z(1), such that y≤χ [a,b] (x)≤z (1.7) and Z 1 0 y(x)−z(x)dx≤ε. (1.8) We can then get that lim N→∞ N P n=1 χ [a,b] (x n ) N ≥ lim N→∞ N P n=1 y(x n ) N = Z 1 0 y(x)dx (direct result from (1.6)) ≥ Z 1 0 z(x)dx−ε (deduced from (1.8)) ≥ Z 1 0 χ [a,b] (x n )dx−ε (deduced from (1.7)) Similarly, we can get that lim N→∞ N P n=1 χ [a,b] (x n ) N ≤ lim N→∞ N P n=1 z(x n ) N = Z 1 0 z(x)dx 9 ≤ Z 1 0 y(x)dx+ε ≤ Z 1 0 χ [a,b] (x n )dx+ε Since ε can be any real number greater than 0, so we get that lim N→∞ N P n=1 χ [a,b] (x n ) N = Z 1 0 χ [a,b] (x)dx =b−a which is directly the definition of u.d mod 1. Therefore,x n is uniformly distributed mod 1 and we show that (iii)⇒ (i) which completes the proof of Weyl’s Criterion. 10 1.3 Example: x n =λn(exercise(2.5) from [2]) Now that we have proved the Weyl’s Criterion, let us apply the Weyl’s Criterion to some simple sequences and see the result. First, we would like to apply to the sequence x n =λn where λ can be any real number. This is also the exercise 2.5 (page 23) from [2]. (a) Ifλ is a rational number, that isλ∈Q, then we can letλ equal to r/m (r∈R, m∈N, HCF(r,m) = 1). Then we see that λn takes only finite distinct values in [0,1). Suppose for n from 0 to q, then we have 0,{ r m },{ 2r m },...,{ qr m } Thus, λn is not uniformly distributed mod 1 with λ being any rational numbers. (b) Suppose that λ is irrational, that is λ∈ I. Since for `∈ Z\{0} we have that e 2πi`λ 6= 1. Then by the formula of the summation of geometric series(S n = a 1 1−q n 1−q , where a 1 is the first term of the series and q is the ratio) and Weyl’s Criterion, we get that N P n=1 e 2πi`λn N = 1 N 1−e 2πi`λN 1−e 2πi`λ Therefore, we get N P n=1 e 2πi`λn N ≤ 1 N 2 |1−e 2πi`λ | As N→∞, 1 N 2 |1−e 2πi`λ | → 0. This means that lim N→∞ N P n=1 e 2πi`λn N = 0 for allλ6= 0. Therefore, λn is uniformly distributed mod 1 with λ being an irrational number. 11 Next, we will see two cases with λ being either rational or irrational by showing their quantile-quantile plots 2 . If the all the points line up in the plot, we can say that it is uniform distribution and vice versa. For irrational case, we take λ = e. For rational case, we take λ = 3 2 Figure 1.3: qq-plot for distribution of en mod 1 2 a quantile-quantile plot compares a distribution to uniform distribution 12 Figure 1.4: qq-plot for distribution of 3 2 n mod 1 By the qq-plots for distributions of en mod 1, we can see that all data points of en mod 1 agree with uniform distribution. Hence, sequence en is uniformly distributed mod 1. However, by the qq-plots for distribution of 3 2 n mod 1, all data points of 3 2 n mod 1 do not agree with uniform distribution. Hence, sequence 3 2 n mod 1 is not uniformly distributed mod 1. 13 1.4 Euler Summation Formula Before starting the analysis of e n , we have to introduce one more important formula. In this section, we introduce the Euler summation formula. Basically, Euler summation formula gives us a formula that connects the summation of a series and the integral of this series. Technologically, we can evaluate the summation of a finite or infinite series by integrals and vice versa. The formula was discovered by Leonhard Euler around 1735. The formula states that if a and b are natural numbers and f(x) is a complex or real valued continuous function for real numbers x in [a,b], then the summation S =f(a+1)+f(a+2)+...+f(b−1)+f(b) (1.9) can be evaluated by the integral (vice versa) F(x) = Z b a f(x)dx (1.10) By Weyl’s Criterion, we are evaluating N P n=1 F(n). We need another form of the Euler summation formula, this form is proved by Tom M.Apostol [6]. The formula states that if F(t) is a complex-valued function with a continuous derivative on 1≤t≤N, where N≥ 1 is an integer, then 3 N X n=1 F(n) = Z N 1 f(t)dt+ 1 2 (F(1)+F(N))+ Z N 1 ({t}− 1 2 )F 0 (t)dt (1.11) 3 {t} was introduced in the definition of u.d mod 1 14 Chapter 2 An analysis of sequence x n =e n In chapter 1, we already showed that for simple function such as x n =en is uniformly distributed mod 1. According to Wikipedia [7], it is not known whether the sequence e n is uniformly distributed mod 1. Hence, in this chapter, we would like to prove whether function like e n is uniformly distributed mod 1 or not. 2.1 Weyl-Koksma’s Metric Theorem To find whether e n is uniformly distributed mod 1 or not, we first take a look on the following theorem that could be helpful for our prediction. Theorem 2.1 (a) Let a> 1 be a real number, then the sequence{λa n } ∞ n=1 is uniformly distributed modulo 1 for almost all real λ (Weyl). (b)Let a be a nonzero real number, then the sequence{λa n } ∞ n=1 is uniformly distributed modulo 1 for almost all real λ> 1 (Koksma). Proof. See [8] and [9]. 2.2 Distribution of sequence e n mod 1 With the Weyl-Koksma’s Metric theorem, we could predict that e n could be uniformly distributed mod 1. Let us take a look at the distribution of e n mod 1 as another prediction. We all know that e n is an increasing function on IR, but what about e n mod 1? Let us take a look at the distribution of e n mod 1. 15 Figure 2.1: distribution of e n mod 1 By the distribution ofe n mod 1, we can figure out that the distribution ofe n mod 1 is random for the interval [0,40]. For interval [40,50], e n mod 1 remains 0. The distribution of e n mod 1 implies that e n may not be uniformly distributed mod 1. 2.3 Quantile-Quantile plot for e n mod 1 Next, we would like to use the quantile-quantile plot to see the distribution of e n mod 1. Below is the graph. 16 Figure 2.2: qq-plot for distribution of e n mod 1 Just like what we did in chapter 1, if all data points agree with uniform distribution, then we would like to say the sequence is uniformly distributed. By the quantile-quantile plot for e n mod 1, we can say that the sequence e n is not uniformly distributed mod 1 since all data points do not agree with uniform distribution. Next, we would like to show that e n is not uniformly distributed mod 1 mathematically. 17 2.4 Combination of Euler summation formula and sequence e n Lemma 2.1 If lim N→∞ N P n=1 e 2πi`xn N 6= 0, then we say that the sequence x n is not uniformly distributed mod 1. proof. direct result from Weyl’s Criterion. Hence, in order to prove that e n is not uniformly distributed mod 1, we need to show lim N→∞ N P n=1 e 2πi`e n N 6= 0 By (1.11), we can write N P n=1 e n as the following: N X n=1 e 2πi`e n = Z N 1 e 2πi`e t dt+ 1 2 (e 2πi`e +e 2πi`e N )+ Z N 1 ({t}− 1 2 )2πi`e t+2πi`e t dt (2.1) (a) We first look at the second term on the right of (2.1). The second term on the right of (2.1), divided by N is 1 2 ( e 2πi`e +e 2πi`e N N ). Note that e 2πi`e N tends to 0 as N →∞. For lim N→∞ e 2πi`e N N , notice that we have lim N→∞ e 2πi`e N N = lim N→∞ cos(2πi`e N )+isin(2πi`e N ) N = 0 Since the numerator has magnitude equals to 1. Hence, the second term does converge to 0. 18 (b) Next we look at the third term on the right side of (2.1). For the third term on the right side of (2.1), We have Z N 1 ({t}− 1 2 )2πi`e t+2πi`e t dt. Note that for ({t}− 1 2 ), since{t} is in the interval [0,1], then ({t}− 1 2 )≤ 1 2 . Therefore, we have that Z N 1 ({t}− 1 2 )2πi`e t+2πi`e t dt≤ 1 2 Z N 1 2πi`e t+2πi`e t dt Let f(x) =e 2πi`e x , then we have d[f(x)] dx = 2πi`e t+2πi`e t . Hence, we can get Z N 1 2πi`e t+2πie t dt =e 2πi`e N −e 2πi`e Divided by N, we can get e 2πi`e N −e 2πi`e N which is similar to the second term on the right side of 2.1. By direct comparison test , if the integral R b a f(x)dx converges and 0≤g(x)≤f(x) for the interval [a,b], then R b a g(x)dx also converges. In our case, let g(x) = ({t}− 1 2 )2πi`e t+2πi`e t and f(x) = 1 2 (2πi`e t+2πi`e t ), then g(x)≤f(x) for the interval [1,N]. Using the result from part (a), we can show that lim N→∞ 1 2 ( e 2πi`e N −e 2πi`e N ) = 0 19 Hence, we conclude that the third term on the right side of (2.1) does converge to 0 as N→∞. (c) For the first term on the right side of (2.1), we have R N 1 e 2πi`e t dt. Let u = 2πi`e t (u∈ [2πi`e,2πi`e N ]), then du = 2πie t dt⇒ du u = dt. Hence, the original integral becomes Z 2πi`e N 2πi`e e u u du To get the integral, we introduce a function called exponential integral. For gen- eral information about the exponential integral, see[11] and [12]. The exponential integral defines a function Ei(x) equal to R ∞ x e t t dt. Hence, we can calculate that Z 2πi`e N 2πi`e e u u du =Ei(2πi`e N )−Ei(2πi`e) Divided by N and we get the first term on the right side of (2.1) to be Ei(2πi`e N )−Ei(2πi`e) N . By figure 2,3 below, we can see the distri- bution of the function Ei(2πi`e x ). From this graph, we can see that lim N→∞ Ei(2πi`e N )−Ei(2πi`e) N = lim N→∞ 3.1416−Ei(2πi`e) N = 0. Hence, the first term on the right side of (2.1) converges to 0 as N→∞. Since all three terms on the right side of (2.1) converges to 0, we would like to say that e n is uniformly distributed mod 1 by Weyl’s Criterion. 2.5 Another approach Based on 2.4, we have presented that e n is uniformly distributed mod 1 by Weyl’s Criterion. This contradicts to the quantile-quantile plot we made in 2.3. Therefore, we would like to use the definition to prove it again. 20 Figure 2.3: absolute value of Ei(2πi`e x ) According to 1.1, if e n is uniformly distributed mod 1, then for every pair a,b of real numbers, we have lim N→∞ A([a,b);N;e n ) N =b−a However, consider a = M, b = M +1, then between log(M) and log(M +1) 1 , there are about log(M + 1)−log(M + 1 2 ) numbers n with e n mod 1 > 1 2 and log(M + 1 2 )−log(M) numbers n with e n mod 1 < 1 2 . Since log(M +1)−log(M + 1 2 ) =log( M +1 M + 1 2 ) 1 log denotes the natural logarithm here 21 and log(M + 1 2 )−log(M) =log( M + 1 2 M ) Hence, these counts differ by a number of log( M +1 M + 1 2 )−log( M + 1 2 M ). Therefore, each element of e n is not equally likely to fall anywhere in its range of interval. Hence,e n is not uniformly distributed mod 1. This is consistent with the quantile- quantile plot we made in 2.3. 22 Chapter 3 Conlusion Although Weyl’s criterion showed e n should be uniformly distributed mod 1, we have presented the bad distribution of the sequence e n by the definition of uniform distribution mod 1. Transcendental numbers with badly distributed powers were proved by David W, Boyd in his paper, see [10]. The bad performance of the distribution of e n can be considered as a direct consequence of very general result in Boyd’s thesis. On the other hand, the proof presented here provides more details for the specific case of a sequence of the form λa n . Our proof may be helpful for finding other transcendental numbers a such that a n is not uniform distributed mod 1. In Boyd’s thesis, there are also general versions of transcendental numbers with badly distributed powers. However, so far, I have not found enough papers and works that are dealing with the related topics. Several topics below deserve further study and research: (1) Find a continuous function f, such that f(n)e n is uniform distributed mod 1. Although e n is not uniform distributed mod 1, multiplying by a function may work. I have tested the sequence sin(n)e n , see figure 3.1. From the quantile-quantile plot for sin(n)e n mod 1, we can see that sin(n)e n is closer to uniform than e n . More closer cases could be presented in the future. 23 Figure 3.1: qq-plot for sin(n)e n mod 1 (2) Find a computable real number a> 1 such that a n mod 1 is uniform distributed mod 1. Although Weyl-Koksma’s Metric Theorem has proved such existence, no explicit example and proof were provided. (3) Find a computable real number a> 1 such that a n mod 1 is not uniform distributed mod 1. For this question, we would basically deal with the sequence ( 3 2 ) n .By the quantile-quantile plot for this sequence in figure 3.2, we can see that the first two-thirds of the sequence in [0,50] agree with uniform distribution. However, the rest part is not close to uniform distribution. Would this be an example to our question? (4) Why does Weyl’s criterion show that e n is uniformly distributed mod 1 while the definition and quantile-quantile plot not? One answer would be the bad distribution of e n . In the quantile-quantile plot, we see that some of the data points agree with uniform distribution. On the other hand, by the definition, although there are difference between counts, However, the difference 24 Figure 3.2: qq-plot for ( 3 2 ) n mod 1 log( M +1 M + 1 2 )−log( M + 1 2 M ) =log( M 2 +M M 2 +M +1 ) tends to be smaller as M becomes larger. Maybe we can call the sequence e n as almost uniformly distributed mod 1. 25 Bibliography [1] Greicius, Tony On Pi Day, How Scientists Use This Number. NASA, March 13, 2015. https://www.nasa.gov/jpl/on-pi-day-how-scientists-use-this-number [2] Kuipers, Lauwerens. and Harald Niederrditer, Uniform distribution of se- quences. Courier Corporation, 2012. [3] Tschinkel, Yuri, Algebraic varieties with many rational points. Arithmetic ge- ometry 8: 243-334, 2008. [4] Zelditch, Steve, Quantum ergodicity and mixing of eigenfunctions. arXiv preprint math-ph/0503026, 2005. [5] Chazelle Bernard, The discrepancy method. International Symposium on Algo- rithm and Computation. Springer, Berlin, Heidelberg, 1998. [6] Apostol Tom, M, An elementary view of Euler’s summation formula. The American Mathematical Monthly 106.5: 409-418, 1999. [7] Wikipedia Equidistributed Sequence. Wikipedia, Wikimedia Foundation, 2018 https ://en.wikipedia.org/wiki/Equidistributed_sequence [8] HermannWeyl,ÜberdieGleichverteilungvonZahlenmoduloEins.Math.Ann., 77, 313-352, 1916. 26 [9] J.F.Koksma, Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins. Compositio Math., 2, 250-258, 1935 [10] Boyd David W, Transcendental numbers with badly distributed powers. Pro- ceedings of the American Mathematical Society 23.2: 424-427, 1969. [11] Harris Frank E, Tables of the exponential integral Ei(x). Mathematical Tables and Other Aids to Computation 11.57: 9-16, 1957. [12] Geller Murray and Edward W. Ng, A table of integrals of the exponential integral.. Journal of Research of the National Bureau of Standards 73.3: 191- 210, 1969. [13] Rudin, Walter. Principles of mathematical analysis. Vol. 3. New York: McGraw-hill, 1964. 27 Appendix A Appendix 1 The value of exponential integral Ei(2πi`e x ) for some specific number are given in here for a supplement. All calculations are direct results from MATLAB. Ei(2πi`e) =−0.0563+3.1564i Ei(2πi`e 2 ) = 0.0142+3.1578i Ei(2πi`e 3 ) =−0.0040+3.1348i Ei(2πi`e 4 ) =−0.0017+3.1440i Ei(2πi`e 5 ) = 0.0006+3.1425i Ei(2πi`e 6 ) = 0.0002+3.1419i Ei(2πi`e 7 ) =−0.0001+3.1417i Ei(2πi`e 8 ) = 0.0000+3.1415i Ei(2πi`e 9 ) = 0.0000+3.1416i Ei(2πi`e 10 ) = 0.0000+3.1416i Ei(2πi`e 11 ) = 0.0000+3.1416i Ei(2πi`e 12 ) = 0.0000+3.1416i Ei(2πi`e 13 ) = 0.0000+3.1416i Ei(2πi`e 14 ) = 0.0000+3.1416i Ei(2πi`e 15 ) = 0.0000+3.1416i Ei(2πi`e 16 ) = 0.0000+3.1416i Ei(2πi`e 17 ) = 0.0000+3.1416i Ei(2πi`e 18 ) = 0.0000+3.1416i 28 Ei(2πi`e 19 ) = 0.0000+3.1416i Ei(2πi`e 20 ) = 0.0000+3.1416i 29 Appendix B Appendix 2 Here are the MATLAB codes used for plots in this thesis. Figure B.1: code for the distribution and qq-plot 30

Abstract (if available)

Abstract Ever since the Weyl criterion was introduced, many mathematicians tried to answer the question: Is there a transcendental number a > 1 such that aⁿ is uniform distribution (mod 1) or not. In 1964, Woody Dudley proved that though cosnθ is not uniformly distributed (mod 1) for almost all θ, f(n)cosnθ is, where f is any function such that f(n) goes to infinity with n, no matter how slowly. ❧ In this thesis, the Weyl criterion is used to figure out a transcendental number a > 1 such that aⁿ is uniform distributed (mod 1) or not. To help find such a transcendental number, a method called Euler summation will be introduced as a key to this problem. The transcendental number e will be used as an example in this thesis.

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Creator Gu, Yuxuan (author)

Core Title Uniform distribution of sequences: Transcendental number and U.D. mod 1

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Applied Mathematics

Publication Date 07/26/2018

Defense Date 06/20/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,transcendental number,uniform distribution of sequence module 1

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Advisor Sacker, Robert (committee chair), Arratia, Richard (committee member), Lototsky, Sergey (committee member)

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