The Arithmetic and Geometry of Salem Numbers
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Introduction to Mahler Measure Universidad De Buenos Aires, Argentina July 23Rd, 2003 Matilde N
Introduction to Mahler Measure Universidad de Buenos Aires, Argentina July 23rd, 2003 Matilde N. Lal´ın{ University of Texas at Austin 1 1. Definition of Mahler Measure and Lehmer's question Looking for large primes, Pierce [18] proposed the following in 1918: Consider P 2 Z[x] monic, and write Y P (x) = (x − αi) i then, we look at Y n ∆n = (αi − 1) i Since the αi are integers and by applying Galois theory, it is easy to see that ∆n 2 Z. n Note that if P = x − 2, we get the sequence ∆n = 2 − 1. The idea is to look for primes among the factors of ∆n. The prime divisors of such integers must satify some congruence conditions that are quite restrictive, hence they are easier to factorize than a randomly given number. Moreover, one can show that ∆mj∆n if mjn. Then we may look at the numbers ∆ p p prime ∆1 In order to minimize the number of trial divisions, the sequence ∆n should grow slowly. Lehmer [15] studied ∆n+1 , observed that ∆n jαn+1 − 1j jαj if jαj > 1 lim = n!1 jαn − 1j 1 if jαj < 1 and suggested the following definition: Definition 1 Given P 2 C[x], such that Y P (x) = a (x − αi) i define the Mahler measure 2 of P as Y M(P ) = jaj maxf1; jαijg (1) i The logarithmic Mahler measure is defined as X + m(P ) = log M(P ) = log jaj + log jαij (2) i [email protected] { http://www.ma.utexas.edu/users/mlalin 2The name Mahler came later after the person who successfully extended this definition to the several- variable case. -
Quartic Salem Numbers Which Are Mahler Measures of Non-Reciprocal 2-Pisot Numbers Tome 32, No 3 (2020), P
Toufik ZAÏMI Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers Tome 32, no 3 (2020), p. 877-889. <http://jtnb.centre-mersenne.org/item?id=JTNB_2020__32_3_877_0> © Société Arithmétique de Bordeaux, 2020, tous droits réservés. L’accès aux articles de la revue « Journal de Théorie des Nom- bres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est con- stitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 32 (2020), 877–889 Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers par Toufik ZAÏMI Résumé. Motivé par une question de M. J. Bertin, on obtient des paramé- trisations des polynômes minimaux des nombres de Salem quartiques, disons α, qui sont des mesures de Mahler des 2 -nombres de Pisot non-réciproques. Cela nous permet de déterminer de tels nombres α, de trace donnée, et de déduire que pour tout entier naturel t (resp. t ≥ 2), il y a un nombre de Salem quartique, de trace t, qui est (resp. qui n’est pas) une mesure de Mahler d’un 2 -nombre de Pisot non-réciproque. -
On the Behavior of Mahler's Measure Under Iteration 1
ON THE BEHAVIOR OF MAHLER'S MEASURE UNDER ITERATION PAUL FILI, LUKAS POTTMEYER, AND MINGMING ZHANG Abstract. For an algebraic number α we denote by M(α) the Mahler measure of α. As M(α) is again an algebraic number (indeed, an algebraic integer), M(·) is a self-map on Q, and therefore defines a dynamical system. The orbit size of α, denoted #OM (α), is the cardinality of the forward orbit of α under M. We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exist algebraic units of larger degree with arbitrarily large but finite orbit size. 1. Introduction The Mahler measure of an algebraic number α with minimal polynomial f(x) = n anx + ··· + a0 2 Z[x] is defined as: n n Y Y M(α) = janj maxf1; jαijg = ±an αi: i=1 i=1 jαij>1 Qn where f(x) = an i=1(x − αi) 2 C[x]. It is clear that M(α) ≥ 1 is a real algebraic integer, and it follows from Kronecker's theorem that M(α) = 1 if and only if α is a root of unity. Moreover, we will freely use the facts that M(α) = M(β) whenever α and β have the same minimal polynomial, and that M(α) = M(α−1). D.H. Lehmer [10] famously asked in 1933 if the Mahler measure for an algebraic number which is not a root of unity can be arbitrarily close to 1. -
The Smallest Perron Numbers 1
MATHEMATICS OF COMPUTATION Volume 79, Number 272, October 2010, Pages 2387–2394 S 0025-5718(10)02345-8 Article electronically published on April 26, 2010 THE SMALLEST PERRON NUMBERS QIANG WU Abstract. A Perron number is a real algebraic integer α of degree d ≥ 2, whose conjugates are αi, such that α>max2≤i≤d |αi|. In this paper we com- pute the smallest Perron numbers of degree d ≤ 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computa- tions use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C 1. Introduction Let α be an algebraic integer of degree d, whose conjugates are α1 = α, α2,...,αd and d d−1 P = X + b1X + ···+ bd−1X + bd, its minimal polynomial. A Perron number, which was defined by Lind [LN1], is a real algebraic integer α of degree d ≥ 2 such that α > max2≤i≤d |αi|.Any Pisot number or Salem number is a Perron number. From the Perron-Frobenius theorem, if A is a nonnegative integral matrix which is aperiodic, i.e. some power of A has strictly positive entries, then its spectral radius α is a Perron number. Lind has proved the converse, that is to say, if α is a Perron number, then there is a nonnegative aperiodic integral matrix whose spectral radius is α.Lind[LN2] has investigated the arithmetic of the Perron numbers. -
The Mahler Measure of Algebraic Numbers: a Survey
THE MAHLER MEASURE OF ALGEBRAIC NUMBERS: A SURVEY CHRIS SMYTH Abstract. A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Re- lated results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed. Some generalisations are given too, though not to Mahler measure of polynomials in more than one variable. 1. Introduction d d Let P (x) = a0z + + ad = a0 i=1(z αi) be a nonconstant polynomial with (at first) complex· · ·coefficients. Then,−following Mahler [101] its Mahler measure is defined to be Q 1 M(P ) := exp log P (e2πit) dt ; (1) j j Z0 the geometric mean of P (z) for z on the unit circle. However M(P ) had appeared earlier in a papj er ofj Lehmer [94], in an alternative form M(P ) = a α : (2) j 0j j ij jαYi|≥1 The equivalence of the two definitions follows immediately from Jensen's for- mula [88] 1 log e2πit α dt = log α : j − j + j j Z0 Here log+ x denotes max(0; log x). If a0 1, then clearly M(P ) 1. This is the case when P has integer coefficienj jts;≥ we assume henceforth≥that P is of this form. Then, from a result of Kronecker [90], M(P ) = 1 occurs only if P is a power of z times a cyclotomic polynomial. In [101] Mahler called M(P ) the measure of the polynomial P , apparently to distinguish it from its (na¨ıve) height. This was first referred to as Mahler's measure by Waldschmidt [165, p.21] in 1979 (`mesure de Mahler'), and soon afterwards by Boyd [33] and Durand [75], in the sense of \the function that Mahler called `measure' ", rather than as a name. -
Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems
Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems Dissertation zur Erlangung des akademischen Grades eines Doktors der Mathematik (Dr. math.) vorgelegt von Chrizaldy Neil Ma~nibo Fakult¨atf¨urMathematik Universit¨atBielefeld April 2019 Gedruckt auf alterungsbest¨andigem Papier ◦◦ISO 9706 1. Berichterstatter: Prof. Dr. Michael Baake Universit¨atBielefeld, Germany 2. Berichterstatter: A/Prof. Dr. Michael Coons University of Newcastle, Australia 3. Berichterstatter: Prof. Dr. Uwe Grimm The Open University, Milton Keynes, UK Datum der m¨undlichen Pr¨ufung:04 Juni 2019 i Contents Acknowledgementsv Introduction viii 1. Prerequisites 1 d 1.1. Point sets in R .....................................1 1.2. Symbolic dynamics and inflation rules........................1 1.2.1. Substitutions..................................1 1.2.2. Perron{Frobenius theory............................3 1.2.3. The symbolic hull...............................4 1.2.4. Inflation systems and the geometric hull...................5 1.3. Harmonic analysis and diffraction...........................7 1.3.1. Fourier transformation of functions......................7 1.3.2. Measures....................................7 1.3.3. Decomposition of positive measures.....................9 1.3.4. Autocorrelation and diffraction measure...................9 1.4. Lyapunov exponents.................................. 11 1.4.1. Lyapunov exponents for sequences of matrices................ 11 1.4.2. Matrix cocycles................................. 12 1.4.3. Ergodic theorems............................... -
A Systematic Construction of Almost Integers Maysum Panju University of Waterloo [email protected]
The Waterloo Mathematics Review 35 A Systematic Construction of Almost Integers Maysum Panju University of Waterloo [email protected] Abstract: Motivated by the search for “almost integers”, we describe the algebraic integers known as Pisot numbers, and explain how they can be used to easily find irrational values that can be arbitrarily close to whole numbers. Some properties of the set of Pisot numbers are briefly discussed, as well as some applications of these numbers to other areas of mathematics. 1 Introduction It is a curious occurrence when an expression that is known to be a non-integer ends up having a value surprisingly close to a whole number. Some examples of this phenomenon include: eπ π = 19.9990999791 ... − 23 5 = 109.0000338701 ... 9 88 ln 89 = 395.0000005364 ... These peculiar numbers are often referred to as “almost integers”, and there are many known examples. Almost integers have attracted considerable interest among recreational mathematicians, who not only try to generate elegant examples, but also try to justify the unusual behaviour of these numbers. In most cases, almost integers exist merely as numerical coincidences, where the value of some expression just happens to be very close to an integer. However, sometimes there actually is a clear, mathematical reason why certain irrational numbers should be very close to whole numbers. In this paper, we’ll look at the a set of numbers called the Pisot numbers, and how they can be used to systematically construct infinitely many examples of almost integers. In Section 2, we will prove a result about powers of roots of polynomials, and use this as motivation to define the Pisot numbers. -
Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds
COUNTING SALEM NUMBERS OF ARITHMETIC HYPERBOLIC 3{ORBIFOLDS MIKHAIL BELOLIPETSKY, MATILDE LAL´IN, PLINIO G. P. MURILLO, AND LOLA THOMPSON Abstract. It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines cQ1=2 +O(Q1=4) square-rootable Salem numbers of degree 4 which are less than or equal to Q. This quantity can be compared to the total number of such Salem numbers, 4 3=2 which is shown to be asymptotic to 3 Q + O(Q). Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic 3-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions 2 and 3. 1. Introduction A Salem number is a real algebraic integer λ > 1 such that all of its Galois conjugates except λ−1 have absolute value equal to 1. Salem numbers appear in many areas of mathematics including algebra, geometry, dynamical systems, and number theory. They are closely related to the celebrated Lehmer's problem about the smallest Mahler measure of a non-cyclotomic polynomial. We refer to [Smy15] for a survey of research on Salem numbers. It has been known for some time that the exponential lengths of the closed geodesics of an arithmetic hyperbolic n-dimensional manifold or orbifold are given by Salem numbers. -
Salem Numbers, Pisot Numbers, Mahler Measure and Graphs
SALEM NUMBERS, PISOT NUMBERS, MAHLER MEASURE AND GRAPHS JAMES MCKEE AND CHRIS SMYTH ABSTRACT. We use graphs to define sets of Salem and Pisot numbers, and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the n-th derived set of the set of Pisot numbers comes from a graph. We define 1 ¤ the Mahler measure of a graph, and find all graphs of Mahler measure less than 2 1 ¡£¢ 5 . Finally, we list all small Salem numbers known to be definable using a graph. 1. INTRODUCTION The work described in this paper arose from the following idea: that one way of studying algebraic integers might be by associating combinatorial objects with them. Here, we try to do this for two particular classes of algebraic integers, Salem numbers and Pisot numbers, the associated combinatorial objects being graphs. We also find all graphs of small Mahler measure. All but one of these measures turns out to be a Salem number. θ A Pisot number is a real algebraic integer ¥ 1, all of whose other Galois conjugates have modulus strictly less than 1. A Salem number is a real algebraic integer τ ¥ 1, whose other conjugates all have modulus at most 1, with at least one having modulus exactly 1. It follows deg P 1 § τ ¦ ¨ § © ¦ § τ that the minimal polynomial P ¦ z of is reciprocal (that is, z P 1 z P z ), that is a τ τ τ τ 1 ¦ § conjugate of , that all conjugates of other than and have modulus exactly 1, and that P z has even degree. -
Mathematical Reviews on the Web C Copyright American Mathematical Society 2004
Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2004 1 997 583 11J68 11R06 Borwein, Peter (3-SFR-MS); Hare, Kevin G. (3-SFR-MS) Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers. (English. English summary) Trans. Amer. Math. Soc. 355 (2003), no. 12, 4767–4779 (electronic). 1 986 822 11D72 11P05 11Y50 Borwein, Peter (3-SFR); Lisonˇek,Petr (3-SFR); Percival, Colin (3-SFR) Computational investigations of the Prouhet-Tarry-Escott problem. (English. English summary) Math. Comp. 72 (2003), no. 244, 2063–2070 (electronic). The Prouhet-Tarry-Escott problem asks for two distinct multisets of integers X = {x1, . , xn} and Y = {y1, . , yn} such that n n X e X e xi = yi i=1 i=1 for e = 1, 2, . , k for some integer k < n. If k = n − 1, such a solution is called ideal. The authors describe a method for searching for ideal symmetric solutions. They make an extensive search for solutions up to n = 12. They find two new solutions for n = 10, which are much smaller than the solutions found by A. Letac in the 1940s. Maurice Mignotte (F-STRAS) 1 986 811 11C08 30C10 Borwein, P. B. (3-SFR-MS); Pinner, C. G. (1-KSS); Pritsker, I. E. (1-OKS) Monic integer Chebyshev problem. (English. English summary) Math. Comp. 72 (2003), no. 244, 1901–1916 (electronic). Results from MathSciNet: Mathematical Reviews on the Web c Copyright American Mathematical Society 2004 2004a:11069 11J72 11J82 33D15 41A21 Borwein, Peter B. (3-SFR-MS); Zhou, Ping [Zhou, Ping2] (3-SFX-MSC) On the irrationality of a certain multivariate q series. -
Small Salem Graphs
Department of Mathematics, Royal Holloway College, University of London July 2016 Small Salem Graphs PhD thesis by Jonathan Charles James Cooley Supervised by Professor James F McKee 1 Declaration of Authorship I Jonathan Charles James Cooley hereby declare that this thesis and the work presented in it is entirely my own. Where I have consulted the work of others, this is always clearly stated. July 2016 2 Abstract The aim for this thesis was to produce the first systematic catalogue of small Salem graphs, and to illustrate and enumerate those with interesting properties. This is done in the central section, chapters 3 to 5. That is preceded by two introductory chapters, the first dealing with definitions and motivation, the second concerning the computational methods used to construct the catalogue. A new isomorphism testing algorithm is presented which has proved highly successful in practice, but an example is constructed in which it is useless. A complete classification of circulant Salem graphs is conjectured. For Salem graphs, their Salem number and their Mahler measure are precisely the same thing, and some observations were made which enabled the completion of the classification of all graphs with Mahler measure below , the very well-known ‘golden number’. The final chapter is an exposition of the paper which reports this completion of the classification. 3 Contents Chapter 1 Definitions and motivation 1.1 Salem numbers 1.2 Totally real algebraic integers, and graphs 1.3 Linking the previous two sections 1.4 Combinatorial objects and polynomials -
Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks
Dynamics on K3 surfaces: Salem numbers and Siegel disks Curtis T. McMullen 19 January, 2001 Abstract This paper presents the first examples of K3 surface automorphisms f : X X with Siegel disks (domains on which f acts by an irrational rotation).→ The set of such examples is countable, and the surface X must be non-projective to carry a Siegel disk. These automorphisms are synthesized from Salem numbers of de- gree 22 and trace 1, which play the role of the leading eigenvalue for f ∗ H2(X). The construction− uses the Torelli theorem, the Atiyah-Bott fixed-point| theorem and results from transcendence theory. Contents 1 Introduction............................ 1 2 K3surfaces ............................ 7 3 AutomorphismsofK3surfaces . 11 4 ErgodicdynamicsonKummersurfaces. 14 5 Siegel disks and transcendence theory . 18 6 HolomorphicLefschetznumbers. 20 7 SiegeldisksonK3surfaces. 22 8 Latticesinnumberfields. 24 9 From Salem numbers to automorphisms . 29 10 ExamplesofSiegeldisks . 30 11 Limits of K¨ahler-Einstein metrics . 34 Research supported in part by the NSF. 2000 Mathematics Subject Classification: 37F50 (11R06, 14J50, 32H50). 1 Introduction The first dynamically interesting automorphisms of compact complex man- ifolds arise on K3 surfaces. Indeed, automorphisms of curves are linear (genus 0 or 1) or of finite order (genus 2 or more). Similarly, automorphisms of most surfaces (includ- ing P2, surfaces of general type and ruled surfaces) are either linear, finite order or skew-products over automorphisms of curves. Only K3 surfaces, Enriques surfaces, complex tori and certain non-minimal rational surfaces admit automorphisms of positive topological entropy [Ca2]. The automor- phisms of tori are linear, and the Enriques examples are double-covered by K3 examples.