BBP-Numbers and Normality
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On the Normality of Numbers
ON THE NORMALITY OF NUMBERS Adrian Belshaw B. Sc., University of British Columbia, 1973 M. A., Princeton University, 1976 A THESIS SUBMITTED 'IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ Adrian Belshaw 2005 SIMON FRASER UNIVERSITY Fall 2005 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Adrian Belshaw Degree: Master of Science Title of Thesis: On the Normality of Numbers Examining Committee: Dr. Ladislav Stacho Chair Dr. Peter Borwein Senior Supervisor Professor of Mathematics Simon Fraser University Dr. Stephen Choi Supervisor Assistant Professor of Mathematics Simon Fraser University Dr. Jason Bell Internal Examiner Assistant Professor of Mathematics Simon Fraser University Date Approved: December 5. 2005 SIMON FRASER ' u~~~snrllbrary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. -
A Theorem on Equidistribution on Compact Groups
Pacific Journal of Mathematics A THEOREM ON EQUIDISTRIBUTION ON COMPACT GROUPS GILBERT HELMBERG Vol. 8, No. 2 April 1958 A THEOREM ON EQUIDISTRIBUTION IN COMPACT GROUPS GILBERT HELMBERG 1. Preliminaries. Throughout the discussions in the following sec- tions, we shall assume that G is a compact topological group whose space is I\ with an identity element e and with Haar-measure μ normal- ized in such a way that μ(G) = l. G has a complete system of inequivalent irreducible unitary representations1 R(λ)(λeA) where β(1) is the identity- (λ) Cλ) representation and rλ is the degree of iϋ . i? (β) will then denote the identity matrix of degree rλ. The concept of equidistribution of a sequence of points was introduced first by H. Weyl [6] for the direct product of circle groups. It has been transferred to compact groups by B. Eckmann [1] and highly generalized by E. Hlawka [4]\ We shall use it in the following from : 1 DEFINITION 1. Let {xv:veω} be a sequence of elements in G and let, for any closed subset M of G, N(M) be the number of elements in the set {xv: xv e M9 v^N}. The sequence {a?v: v e ω] is said to be equidίstributed in G if (1) for all closed subsets M of G, whose boundaries have measured 0. It is easy to see that a sequence which is equidistributed in G is also dense in G. As Eckmann has shown for compact groups with a countable base, and E. Hlawka for compact groups in general, the equidistribution of a sequence in G can be stated by means of the system {R^ΆeΛ} of representations of G. -
Bernoulli Decompositions and Applications
Equidistributed sequences Bernoulli systems Sinai's factor theorem Bernoulli decompositions and applications Han Yu University of St Andrews A day in October Equidistributed sequences Bernoulli systems Sinai's factor theorem Outline Equidistributed sequences Bernoulli systems Sinai's factor theorem It is enough to check the above result for each interval with rational end points. It is also enough to replace the indicator function with a countable dense family of continuous functions in C([0; 1]): Equidistributed sequences Bernoulli systems Sinai's factor theorem A reminder Let fxngn≥1 be a sequence in [0; 1]: It is equidistributed with respect to the Lebesgue measure λ if for each (open or close or whatever) interval I ⊂ [0; 1] we have the following result, N 1 X lim 1I (xn) = λ(I ): N!1 N n=1 It is also enough to replace the indicator function with a countable dense family of continuous functions in C([0; 1]): Equidistributed sequences Bernoulli systems Sinai's factor theorem A reminder Let fxngn≥1 be a sequence in [0; 1]: It is equidistributed with respect to the Lebesgue measure λ if for each (open or close or whatever) interval I ⊂ [0; 1] we have the following result, N 1 X lim 1I (xn) = λ(I ): N!1 N n=1 It is enough to check the above result for each interval with rational end points. Equidistributed sequences Bernoulli systems Sinai's factor theorem A reminder Let fxngn≥1 be a sequence in [0; 1]: It is equidistributed with respect to the Lebesgue measure λ if for each (open or close or whatever) interval I ⊂ [0; 1] we have the following result, N 1 X lim 1I (xn) = λ(I ): N!1 N n=1 It is enough to check the above result for each interval with rational end points. -
Construction of Normal Numbers with Respect to Generalized Lüroth Series from Equidistributed Sequences
CONSTRUCTION OF NORMAL NUMBERS WITH RESPECT TO GENERALIZED LÜROTH SERIES FROM EQUIDISTRIBUTED SEQUENCES MAX AEHLE AND MATTHIAS PAULSEN Abstract. Generalized Lüroth series generalize b-adic representations as well as Lüroth series. Almost all real numbers are normal, but it is not easy to construct one. In this paper, a new construction of normal numbers with respect to Generalized Lüroth Series (including those with an infinite digit set) is given. Our method concatenates the beginnings of the expansions of an arbitrary equidistributed sequence. 1. Introduction In 1909, Borel proved that the b-adic representations of almost all real numbers are normal for any integer base b. However, only a few simple examples for normal numbers are known, like the famous Champernowne constant [Cha33] 0 . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .... Other constants as π or e are conjectured to be normal. The concept of normality naturally extends to more general number expansions, including Generalized√ Lüroth 1 Series [DK02, pp. 41–50]. For example, the numbers 1/e and 2 3 are not normal with respect to the classical Lüroth expansion that was introduced in [Lü83]. In his Bachelor thesis, Boks [Bok09] proposed an algorithm that yields a normal number with respect to the Lüroth expansion, but he did not complete his proof of normality. Some years later, Madritsch and Mance [MM14] transferred Champer- nowne’s construction to any invariant probability measure. However, their method is rather complicated and does not reflect that digit sequences represent numbers. Vandehey [Van14] provided a much simpler construction, but for finite digit sets only, in particular, not for the original Lüroth expansion. -
Random Generators and Normal Numbers
Random Generators and Normal Numbers David H. Bailey1 and Richard E. Crandall2 20 March 2003 Abstract Pursuant to the authors’ previous chaotic-dynamical model for random digits of fun- damental constants [5], we investigate a complementary, statistical picture in which pseu- dorandom number generators (PRNGs) are central. Some rigorous results are achieved: mi ni We establish b-normality for constants of the form i 1/(b c ) for certain sequences (mi), (ni) of integers. This work unifies and extends previously known classes of explicit n normals. We prove that for coprime b, c > 1 the constant αb,c = n=c,c2,c3,... 1/(nb )is b-normal, thus generalizing the Stoneham class of normals [49]. Our approach also re- proves b-normality for the Korobov class [34] βb,c,d, for which the summation index n 2 3 above runs instead over powers cd,cd ,cd ,... with d>1. Eventually we describe an un- countable class of explicit normals that succumb to the PRNG approach. Numbers of the α, β classes share with fundamental constants such as π, log 2 the property that isolated digits can be directly calculated, but for these new classes such computation tends to be surprisingly rapid. For example, we find that the googol-th (i.e. 10100-th) binary bit of α2,3 is 0. We also present a collection of other results—such as digit-density results and irrationality proofs based on PRNG ideas—for various special numbers. AMS 2000 subject classification numbers: 11K16, 11K06, 11J81, 11K45 Keywords: normal numbers, transcendental numbers, pseudo-random number generators 1Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA, [email protected]. -
International Journal of Pure and Applied Mathematics ————————————————————————– Volume 72 No
International Journal of Pure and Applied Mathematics ————————————————————————– Volume 72 No. 4 2011, 515-526 UNIFORM EQUIPARTITION TEST BOUNDS FOR MULTIPLY SEQUENCES Marco Pollanen Department of Mathematics Trent University Peterborough, ON K9J 7B8, CANADA Abstract: For almost all x0 ∈ [0, 1), the multiply sequence xn = axn−1 mod 1, with a > 1 an integer, is equidistributed. In this paper we show that equidis- tributed multiply sequences are not m-equipartitioned for any m> 2. We also provide uniform asymptotic bounds for equipartition tests for such sequences. AMS Subject Classification: 11K06, 11J71 Key Words: multiply sequences, equipartition, ∞-distribution 1. Introduction The sequence xn = axn−1 + c mod 1, where a is a non-negative integer, is an important sequence that arises in number theory, fractals, and applied mathe- matics [4]. A sequence hyni is equidistributed in [0, 1) if for all 0 ≤ a < b ≤ 1, 1 lim χ[a,b)(yn)= b − a, N→∞ N 1≤Xn≤N where χA is the characteristic function of a set A. For the case when a = 1, the sequence hxni was studied by Weyl [7] and shown to be equidistributed if and only if c is irrational. In the case when a> 1, the sequence is referred to as a multiply sequence. It was first shown by Borel that, for a> 1 and c = 0, the sequence is equidistributed for almost all x0. Received: July 6, 2011 c 2011 Academic Publications, Ltd. 516 M. Pollanen As in [2], let hSni be a sequence of propositions about the sequence hyni. We define 1 P (hSni) = lim 1 , N→∞ N Sn isX true 1≤n≤N when the limit exists. -
Mathematical Constants and Sequences
Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) .. -
Sato-Tate Distributions
SATO-TATE DISTRIBUTIONS ANDREW V.SUTHERLAND ABSTRACT. In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato- Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures≤ presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry. 1. AN INTRODUCTION TO SATO-TATE DISTRIBUTIONS Before discussing the Sato-Tate conjecture and Sato-Tate distributions in the context of abelian vari- eties, let us first consider the more familiar setting of Artin motives (varieties of dimension zero). 1.1. A first example. Let f Z[x] be a squarefree polynomial of degree d. For each prime p, let 2 fp (Z=pZ)[x] Fp[x] denote the reduction of f modulo p, and define 2 ' Nf (p) := # x Fp : fp(x) = 0 , f 2 g which we note is an integer between 0 and d. We would like to understand how Nf (p) varies with p. 3 The table below shows the values of Nf (p) when f (x) = x x + 1 for primes p 60: − ≤ p : 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Nf (p) 00111011200101013 There does not appear to be any obvious pattern (and we should know not to expect one, because the Galois group of f is nonabelian). -
List of Numbers
List of numbers This is a list of articles aboutnumbers (not about numerals). Contents Rational numbers Natural numbers Powers of ten (scientific notation) Integers Notable integers Named numbers Prime numbers Highly composite numbers Perfect numbers Cardinal numbers Small numbers English names for powers of 10 SI prefixes for powers of 10 Fractional numbers Irrational and suspected irrational numbers Algebraic numbers Transcendental numbers Suspected transcendentals Numbers not known with high precision Hypercomplex numbers Algebraic complex numbers Other hypercomplex numbers Transfinite numbers Numbers representing measured quantities Numbers representing physical quantities Numbers without specific values See also Notes Further reading External links Rational numbers A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode ℚ);[2] it was thus denoted in 1895 byGiuseppe Peano after quoziente, Italian for "quotient". Natural numbers Natural numbers are those used for counting (as in "there are six (6) coins on the table") and ordering (as in "this is the third (3rd) largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are -
On the Random Character of Fundamental Constant Expansions
On the Random Character of Fundamental Constant Expansions David H. Bailey1 and Richard E. Crandall2 03 October 2000 Abstract We propose a theory to explain random behavior for the digits in the expansions of fundamental mathematical constants. At the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base-2 normality—namely bit randomness in a specific technical sense—for a collection of celebrated constants, including π, log 2,ζ(3), and others. Also on the hypothesis, the number ζ(5)iseitherrationalornormaltobase 2. We indicate a research connection between our dynamical model and the theory of pseudorandom number generators. 1Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA, [email protected]. Bailey’s work is supported by the Director, Office of Computational and Technology Research, Division of Math- ematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC03-76SF00098. 2Center for Advanced Computation, Reed College, Portland, OR 97202 USA, [email protected]. 1 1. Introduction It is of course a long-standing open question whether the digits of π and various other fundamental constants are “random” in an appropriate statistical sense. Informally speaking, we say that a number α isnormaltobaseb if every sequence of k consecutive digits in the base-b expansion of α appears with limiting probability b−k. In other words, if a constant is normal to base 10, then its decimal expansion would exhibit a “7” one- tenth of the time, the string “37” one one-hundredth of the time, and so on. -
BENFORD's LAW, VALUES of L-FUNCTIONS and the 3X + 1
BENFORD'S LAW, VALUES OF L-FUNCTIONS AND THE 3x + 1 PROBLEM ALEX V. KONTOROVICH AND STEVEN J. MILLER Abstract. We show the leading digits of a variety of systems satisfying certain conditions follow Benford's Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, speci¯c to the system. The other is a general technique of applying Poisson Summation to the limiting distribution. We show the distribution of values of L-functions near the central line and (in some sense) the iterates of the 3x + 1 Problem are Benford. 1. Introduction While looking through tables of logarithms in the late 1800s, Newcomb [New] noticed a surprising fact: certain pages were signi¯cantly more worn than others. People were referencing numbers whose logarithm started with 1 more frequently than other digits. In 1938 Benford [Ben] observed the same digit bias in a wide variety of phenomena. Instead of observing one-ninth (about 11%) of entries having a leading digit of 1, as one would expect if the digits 1; 2;:::; 9 were equally likely, over 30% of the entries had leading digit 1 and about 70% had leading digit less than 5. Since log10 2 ¼ 0:301 and log10 5 ¼ 0:699, one may speculate that the probability of observing a digit less than k is log10 k, meaning³ that´ the probability of seeing a particular digit j is log10 (j + 1) ¡ 1 log10 j = log10 1 + j . This logarithmic phenomenon became known as Benford's Law after his paper containing extensive empirical evidence of this distribution in diverse data sets gained popularity. -
A Structure Theorem for Multiplicative Functions Over the Gaussian Integers and Applications
A STRUCTURE THEOREM FOR MULTIPLICATIVE FUNCTIONS OVER THE GAUSSIAN INTEGERS AND APPLICATIONS WENBO SUN Abstract. We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U3-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x2 − y2 = n2 for some Gaussian integer n. The analog of this statement over Z remains open. 1. Introduction 1.1. Structure theory in the finite setting. The structure theorem for functions on Zd is an important tool in additive combinatorics. It has been studied extensively in [4], [8], [9], [10], [11], [14], [17], [24] and [25]. Roughly speaking, the structure theorem says that every function f can be decomposed into one part with a good uniformity property, meaning it has a small Gowers norm, and another with a good structure, meaning it is a nilsequence with bounded complexity. A natural question to ask is: can we get a better decomposition for functions f satis- fying special conditions? For example, Green, Tao and Ziegler ( [11], [14], [17]) gave a refined decomposition result for the von Mangoldt function Λ. They showed that under some modification, one can take the structured part to be the constant 1.