The Fibonacci Word Fractal Alexis Monnerot-Dumaine
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Modular Hyperbolas and Beatty Sequences
MODULAR HYPERBOLAS AND BEATTY SEQUENCES MARC TECHNAU Abstract. Bounds for max{m, m˜ } subject to m, m˜ ∈ Z ∩ [1,p), p prime, z indivisible by p, mm˜ ≡ z mod p and m belonging to some fixed Beatty sequence {⌊nα+β⌋ : n ∈ N} are obtained, assuming certain conditions on α. The proof uses a method due to Banks and Shparlinski. As an intermediate step, bounds for the discrete periodic autocorrelation of the finite sequence 0, ep(y1), ep(y2),..., ep(y(p − 1)) on average are obtained, where ep(t) = exp(2πit/p) and mm ≡ 1 mod p. The latter is accomplished by adapting a method due to Kloosterman. 1. Introduction Consider the modular hyperbola (1.1) H = (m, m˜ ) Z2 [1,p)2 : mm˜ z mod p (with p ∤ z), z mod p { ∈ ∩ ≡ } where the letter p denotes a prime here and throughout. The distribution of points on these modular hyperbolas has attracted wide interest and the interested reader is referred to [28] for a survey of questions related to this topic and various applications. Our particular starting point is the following intriguing property of such hyperbolas: Theorem 1.1. For any prime p and z coprime to p there is always a point (m, m˜ ) ∈ H with max m, m˜ 2(log p)p3/4. z mod p { }≤ Proof. See, e.g., [17, p. 382].1 Loosely speaking, the above theorem states that, when sampling enough points (1, ?), (2, ?′), (3, ?′′), ... H , ∈ z mod p at least one amongst the second coordinates will not be too large. Here we shall investigate analogous distribution phenomena when one of the coordi- nates is required to belong to some fixed Beatty set. -
Properties and Generalizations of the Fibonacci Word Fractal Exploring Fractal Curves José L
The Mathematica® Journal Properties and Generalizations of the Fibonacci Word Fractal Exploring Fractal Curves José L. Ramírez Gustavo N. Rubiano This article implements some combinatorial properties of the Fibonacci word and generalizations that can be generated from the iteration of a morphism between languages. Some graphic properties of the fractal curve are associated with these words; the curves can be generated from drawing rules similar to those used in L-systems. Simple changes to the programs generate other interesting curves. ‡ 1. Introduction The infinite Fibonacci word, f = 0 100 101 001 001 010 010 100 100 101 ... is certainly one of the most studied words in the field of combinatorics on words [1–4]. It is the archetype of a Sturmian word [5]. The word f can be associated with a fractal curve with combinatorial properties [6–7]. This article implements Mathematica programs to generate curves from f and a set of drawing rules. These rules are similar to those used in L-systems. The outline of this article is as follows. Section 2 recalls some definitions and ideas of combinatorics on words. Section 3 introduces the Fibonacci word, its fractal curve, and a family of words whose limit is the Fibonacci word fractal. Finally, Section 4 generalizes the Fibonacci word and its Fibonacci word fractal. The Mathematica Journal 16 © 2014 Wolfram Media, Inc. 2 José L. Ramírez and Gustavo N. Rubiano ‡ 2. Definitions and Notation The terminology and notation are mainly those of [5] and [8]. Let S be a finite alphabet, whose elements are called symbols. A word over S is a finite sequence of symbols from S. -
Monthly Problem 3173, Samuel Beatty, and 1P+1Q=1
1 1 Monthly Problem 3173, Samuel Beatty, and + = 1 p q Ezra Brown Virginia Tech MD/DC/VA Spring Section Meeting Montgomery College April 16, 2016 Brown Beatty Sequences What to expect An interesting pair of sequences Problem 3173 of the American Mathematical Monthly Beatty sequences Samuel Beatty The Proof Beatty sequences could be called Wythoff sequences ::: ::: or even Rayleigh sequences. 1 1 Another appearance of 1 ::: coincidence? p q + = Brown Beatty Sequences An interesting pair of sequences Let p 1 5 2 φ and q φ φ 1 . √ Let A = ( np+ n)~ 1=; 2; 3;::: =and~(B −nq) n 1; 2; 3;::: . Then = {⌊ ⌋ ∶ =A 1; 3; 4;}6; 8; 9; 11{⌊; 12;⌋14∶ ; 16= ; 17; 19;:::} ; and ={ } B 2; 5; 7; 10; 13; 15; 18; 20;::: Looks like the two sequences={ will contain all the} positive integers without repetition. 1 1 It also happens that 1: p q Coincidence? + = Brown Beatty Sequences The Monthly, vol. 33, #3 (March 1926), p. 159 3173. Proposed by Samuel Beatty, University of Toronto. If x is a positive irrational number and y is its reciprocal, prove that the sequences 1 x ; 2 1 x ; 3 1 x ;::: and 1 y ; 2 1 y ; 3 1 y ;::: ( + ) ( + ) ( + ) contain one and only one( number+ ) ( between+ ) ( each+ ) pair of consecutive positive integers. Equivalent statement: If x is a positive irrational number and y is its reciprocal, prove that the sequences 1 x ; 2 1 x ; 3 1 x ;::: and 1 y ; 2 1 y ; 3 1 y ;::: ⌊( + )⌋ ⌊ ( + )⌋ ⌊ ( + )⌋ contain each positive⌊( integer+ )⌋ once⌊ ( without+ )⌋ ⌊ duplication.( + )⌋ Brown Beatty Sequences Problem 3173 rephrased Observation: if p 1 -
Here the Base Β Is Real and the Alphabet Is Subset a of Z
Numeration 2018 22-25 May 2018 Paris France Table of contents Invited talks (Edita Pelantov´a,Robert F. Tichy)3 Establishing and maintaining databases of self-affine tiles, Bandt Christoph [et al.] 1 On the beta-expansion of integers of Z[beta]; connections with self similar tilings, Bertrand-Mathis Anne3 Integer multipliers of real polynomials without nonnegative roots, Brunotte Horst 4 On the magic of some families of fractal dendrites, Cristea Ligia Loretta5 Multiplicative dependence of shifted algebraic numbers, Dubickas Arturas7 Representations of palindromes in the Fibonacci word, Frid Anna9 Random homogeneous beta-expansions and self-similar measures, Hare Kathryn [et al.] 13 Totally positive quadratic integers and numeration, Hejda Tomas [et al.] 14 Toolset for supporting the number system research, Hudoba Peter [et al.] 19 Characterization of rational matrices that admit finite digit representations, Jankauskas Jonas [et al.] 22 1 Unique expansions on fat Sierpinski gaskets, Kong Derong [et al.] 23 Multi-base Representations and their Minimal Hamming Weight, Krenn Daniel [et al.] 24 Infinite families of number systems, Kr´asensk´yJakub [et al.] 26 Ito alpha-continued fractions and matching, Langeveld Niels [et al.] 29 The sum-of-digits function of linearly recurrent number systems and almost primes, Madritsch Manfred 31 Constructing invariant densities for random systems, Maggioni Marta [et al.] 33 Some complexity results in the theory of normal numbers, Airey Dylan [et al.] 36 Mobius Orthogonality for automatic sequences and beyond, -
Beatty Sequences, Fibonacci Numbers, and the Golden Ratio
BEATTY SEQUENCES, FIBONACCI NUMBERS, AND THE GOLDEN RATIO VINCENT RUSSO AND LOREN SCHWIEBERT Abstract. 2 (bnφc)n≥1 and nφ n≥1 are well-known complementary Beatty sequences. An infinite set of complementary Beatty sequences, based on a generalization of ratios of Fibonacci numbers and higher powers of φ, is proved. An open problem posed by Clark Kimberling, the Swappage Problem, is resolved in the affirmative as a special case of this set of complementary Beatty sequences. 1. Introduction A Beatty sequence [9, 1, 10] is generated by an irrational α> 1 as follows: (bαc, b2αc, b3αc,...) = (bnαc)n≥1. Letting β be the number satisfying 1/α + 1/β = 1, the sequences (bnαc)n≥1 and (bnβc)n≥1 are complementary Beatty sequences. See for example [6]. It is well-known that if α = φ, where φ denotes the golden ratio, then β = α + 1. The corresponding sequence (bnαc)n≥1 = (1, 3, 4, 6,...), is the lower Wythoff sequence [7]. The complementary Beatty sequence, (bnβc)n≥1 = (2, 5, 7, 10,...), is the upper Wythoff sequence [7]. 2. Beatty Sequences As noted above, when α = φ, we obtain β = α + 1. From the identity φ2 = φ + 1, we may rewrite β as β = φ2. These complementary Beatty sequences form the basis for Theorem 2.2, which uses the following well-known result. Lemma 2.1. k+1 φ = Fk+1φ + Fk. Proof. This is easily proved by induction. For the base case, observe that F1 = F2 = 1 and φ2 = φ + 1. For the inductive case, assume equality holds for all k<n. -
Pattern, the Visual Mathematics: a Search for Logical Connections in Design Through Mathematics, Science, and Multicultural Arts
Rochester Institute of Technology RIT Scholar Works Theses 6-1-1996 Pattern, the visual mathematics: A search for logical connections in design through mathematics, science, and multicultural arts Seung-eun Lee Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Lee, Seung-eun, "Pattern, the visual mathematics: A search for logical connections in design through mathematics, science, and multicultural arts" (1996). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. majWiif." $0-J.zn jtiJ jt if it a mtrc'r icjened thts (waefii i: Tkirr. afaidiid to bz i d ! -Lrtsud a i-trio. of hw /'' nvwte' mfratWf trawwi tpf .w/ Jwitfl- ''','-''' ftatsTB liovi 'itcj^uitt <t. dt feu. Tfui ;: tti w/ik'i w cWii.' cental cj lniL-rital. p-siod. [Jewy anJ ./tr.-.MTij of'irttion. fan- 'mo' that. I: wo!'.. PicrdoTrKwe, wiifi id* .or.if'iifer -now r't.t/ry summary Jfem2>/x\, i^zA 'jX7ipu.lv z&ietQStd (Xtrma iKc'i-J: ffocub. The (i^piicaiom o". crWx< M Wf flrtu't. fifesspwn, Jiii*iaiin^ jfli dtfifi,tmi jpivxei u3ili'-rCi luvjtv-tzxi patcrr? ;i t. drifters vcy. Symmetr, yo The theoretical concepts of symmetry deal with group theory and figure transformations. Figure transformations, or symmetry operations refer to the movement and repetition of an one-, two , and three-dimensional space. (f_8= I lhowtuo irwti) familiar iltjpg . -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library -
Combinatorics, Automata and Number Theory CANT
Combinatorics, Automata and Number Theory CANT Edited by Val´erie Berth´e LIRMM - Universit´e Montpelier II - CNRS UMR 5506 161 rue Ada, F-34392 Montpellier Cedex 5, France Michel Rigo Universit´e de Li`ege, Institut de Math´ematiques Grande Traverse 12 (B 37), B-4000 Li`ege, Belgium 8 Transcendence and Diophantine approximation Boris Adamczewski CNRS, Universit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France Yann Bugeaud IRMA - Universit´e de Strasbourg-Math´ematiques - CNRS UMR 7501 7 rue Ren´e Descartes, F-67084 Strasbourg cedex, France. The aim of this chapter is to present several number-theoretic problems that reveal a fruitful interplay between combinatorics on words and Diophantine approximation. Finite and infinite words occur naturally in Diophantine approximation when we consider the expansion of a real number in an inte- ger base b or its continued fraction expansion. Conversely, with an infinite word a on the finite alphabet 0, 1, . , b 1 we associate the real number { − } ξa whose base-b expansion is given by a. As well, with an infinite word a on the infinite alphabet 1, 2, 3, . , we associate the real number ζ whose { } a continued fraction expansion is given by a. It turns out that, if the word a enjoys certain combinatorial properties involving repetitive or symmetric patterns, then this gives interesting information on the arithmetical nature and on the Diophantine properties of the real numbers ξa and ζa. We illustrate our results by considering the real numbers associated with two classical infinite words, the Thue-Morse word and the Fibonacci word, see Example 1.2.21 and 1.2.22. -
The Fractional Part of N+ Ø and Beatty Sequences
JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX TAKAO KOMATSU The fractional part of nq + and Beatty sequences Journal de Théorie des Nombres de Bordeaux, tome 7, no 2 (1995), p. 387-406 <http://www.numdam.org/item?id=JTNB_1995__7_2_387_0> © Université Bordeaux 1, 1995, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi- tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti- lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 387 The fractional part of n03B8 + ~ and Beatty sequences par TAKAO KOMATSU 1. Introduction Let () be real and be the denominator of the n-th convergent of the continued fraction expansion of 0. Thus when 0 is rational, 8 = for some integer N. Denote the fractional part of 0 by {9}, the floor, that is integer part, of 0 by and its distance from the nearest integer by ~~8~~. It is well known that the continued fraction convergents are best approx- imations to 0 in the sense that The corresponding results for one-sided best approximations are readily ° derived, see for example van Ravenstein [9j, and are ’ and The purpose of this paper is to give similar results in the inhomogeneous case x8 -- , where 0 and ø are real numbers, not necessarily irrational, and to apply these results to find the characteristic word of the Beatty sequence. -
Prime Numbers with Beatty Sequences
Prime numbers with Beatty sequences William D. Banks Department of Mathematics University of Missouri Columbia, MO 65211 USA [email protected] Igor E. Shparlinski Department of Computing Macquarie University Sydney, NSW 2109, Australia [email protected] September 6, 2011 1 Abstract A study of certain Hamiltonian systems has lead Y. Long to con- jecture the existence of infinitely many primes which are not of the form p = 2 ⌊αn⌋ + 1, where 1 <α< 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately imply that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes p = q ⌊αn + β⌋+a with n 6 N, where α, β are real numbers such that α is positive and irrational of finite type (which is true for almost all α) and a, q are integers with 0 6 a<q 6 N κ and gcd(a, q) = 1, where κ > 0 depends only on α. We also prove a similar result for primes p = ⌊αn + β⌋ such that p ≡ a (mod q). MSC Numbers: 11B83, 11L03, 11N13. Keywords: Beatty sequences, prime numbers, exponential sums. 2 1 Introduction For two fixed real numbers α and β, the corresponding non-homogeneous Beatty sequence is the sequence of integers defined by ∞ Bα,β =(⌊αn + β⌋)n=1 . Beatty sequences appear in a variety of apparently unrelated mathematical settings, and because of their versatility, many arithmetic properties of these sequences have been explored in the literature, such as • structural properties (see [14, 20, 21, 30, 38]); • additive properties (see [5, 24]); • the distribution of arithmetic functions (see [1, 2, 8, 11, 12, 16, 29]); • the distribution of quadratic nonresidues and primitive roots (see [4, 6, 7, 9, 15, 31, 32, 33]). -
On the K-Fibonacci Words
Acta Univ. Sapientiae, Informatica, 5, 2 (2013) 212–226 DOI: 10.2478/ausi-2014-0011 On the k-Fibonacci words Jos´eL.RAM´IREZ Gustavo N. RUBIANO Instituto de Matem´aticas y sus Departamento de Matem´aticas Aplicaciones Universidad Nacional de Colombia, Universidad Sergio Arboleda, Colombia Colombia email: email: [email protected] [email protected] Abstract. In this paper we define the k-Fibonacci words in analogy with the definition of the k-Fibonacci numbers. We study their properties and we associate to this family of words a family of curves with interesting patterns. 1 Introduction Fibonacci numbers and their generalizations have many interesting properties and applications to almost every fields of science and arts (e.g. see [13]). The Fibonacci numbers Fn are the terms of the sequence 0, 1, 1, 2, 3, 5, . wherein each term is the sum of the two previous terms, beginning with the values F0 = 0,andF1 = 1. Besides the usual Fibonacci numbers many kinds of generalizations of these numbers have been presented in the literature. In particular, a generalization is the k-Fibonacci numbers [11]. For any positive real number k,thek-Fibonacci sequence, say {Fk,n}n∈N is defined recurrently by Fk,0 = 0, Fk,1 = 1 and Fk,n+1 = kFk,n + Fk,n−1,n 1. (1) Computing Classification System 1998: I.3.7, G.2.0 Mathematics Subject Classification 2010: 11B39, 05A05 , 68R15 Key words and phrases: Fibonacci word, k-Fibonacci numbers, k-Fibonacci words, k- Fibonacci curves 212 On the k-Fibonacci words 213 In [11], k-Fibonacci numbers were found by studying the recursive applica- tion of two geometrical transformations used in the four-triangle longest-edge (4TLE) partition. -
Beatty Sequences, Continued Fractions, and Certain Shift Operators
Canad. Math. Bull. Vol. 19 (4), 1976 BEATTY SEQUENCES, CONTINUED FRACTIONS, AND CERTAIN SHIFT OPERATORS BY KENNETH B. STOLARSKY ABSTRACT. Let 0 = 0(k) be the positive root of d2 + (k-2)d- k = 0. Let f(n) = [(n + l)d]-[n$] for positive integers n, where [JC] denotes the greatest integer in x. Then the elements of the infinite sequence (/(l),/(2),/(3),...) can be rapidly generated from the finite sequence (/(l), /(2),..., f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nd] itself in this manner. 1. Introduction. Since the "golden mean" «=1(1 + 75) has the expansion l+T^rïT' • * as a simple continued fraction, it is readily associated with the Fibonacci sequence 1,1, 2, 3, 5, 8,.... We shall show there is an apparently very different way in which a is associated with this sequence. Let 101010- • • be the sequence consisting of the 2 numbers 0,1 repeated infinitely often (in that order). If we repeat the first 3 numbers of this sequence infinitely often we get 101101101- • • . If we similarly repeat the first 5 numbers here, we get 1011010110- •• ; for 8 we get 1011010110110101 • • •. It is clear that this process "converges" (see §2 for a precise definition) to a sequence a1a2a3> • • • where a* is 0 or 1. THEOREM 1. Let [JC] denote the largest integer not exceeding x. Then an = l if n = [ma] for some integer m, and an = 0 otherwise. We shall find a similar phenomenon (Theorem 2 of §3) for an infinite class of quadratic irrationalities, and show that it is in fact a consequence of well known properties of simple continued fractions.