ON REAL PARTS OF POWERS OF COMPLEX PISOT NUMBERS

2016 ◽  
Vol 94 (2) ◽  
pp. 245-253 ◽  
Author(s):  
TOUFIK ZAÏMI
Keyword(s):  
Finite Number ◽  
Algebraic Number ◽  
Complex Number ◽  
Convergent Sequence ◽  
Pisot Number ◽  
Pisot Numbers ◽  
Limit Points ◽  
Nonzero Complex Number

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

scholarly journals On the lines passing through two conjugates of a Salem number

2008 ◽  
Vol 144 (1) ◽  
pp. 29-37 ◽  
Author(s):  
ARTŪRAS DUBICKAS ◽  
CHRIS SMYTH
Keyword(s):  
Algebraic Number ◽  
General Position ◽  
Imaginary Axis ◽  
Pisot Number ◽  
Pisot Numbers ◽  
Parallel Lines ◽  
Salem Number ◽  
Pass Through

AbstractWe show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d ≥ 3 is at most [d2/2]-d+2, its conjugates being in general position if this number is attained. If, for instance, d ≥ 4 is even, then the conjugates of α ∈ $\overline{\Q}$ of degree d are in general position if and only if α has 2 real conjugates, d-2 complex conjugates, no three distinct conjugates of α lie on a line and any two lines that pass through two distinct conjugates of α are non-parallel, except for d/2-1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation α1+α2=α3+α4 in distinct conjugates of a Pisot number. The Pisot number $\al_1\,{=}\,(1+\sqrt{3+2\sqrt{5}})/2$ shows that this equation has such a solution.

scholarly journals On the conjugates of certain algebraic integers

2016 ◽  
Vol 99 (113) ◽  
pp. 281-285 ◽  
Author(s):  
Toufik Zaïmi
Keyword(s):  
Natural Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Pisot Number ◽  
Algebraic Integers ◽  
Pisot Numbers

A well-known theorem, due to C. J. Smyth, asserts that two conjugates of a Pisot number, having the same modulus are necessary complex conjugates. We show that this result remains true for K-Pisot numbers, where K is a real algebraic number field. Also, we prove that a j-Pisot number, where j is a natural number, can not have more than 2j conjugates with the same modulus.

scholarly journals On purely periodic beta-expansions of Pisot numbers

2002 ◽  
Vol 166 ◽  
pp. 183-207 ◽  
Author(s):  
Yuki Sano
Keyword(s):  
Natural Extension ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Pisot Number ◽  
Fractal Boundary ◽  
Pisot Numbers ◽  
Dimensional Domain

AbstractWe characterize numbers having purely periodic β-expansions where β is a Pisot number satisfying a certain irreducible polynomial. The main tool of the proof is to construct a natural extension on a d-dimensional domain with a fractal boundary.

scholarly journals ON THE SPECTRA OF PISOT NUMBERS

2011 ◽  
Vol 54 (1) ◽  
pp. 127-132 ◽  
Author(s):  
TOUFIK ZAIMI
Keyword(s):  
Real Number ◽  
Fractional Part ◽  
Pisot Number ◽  
Real Numbers ◽  
Pisot Numbers

AbstractLet θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ ℕ. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all$\alpha \in \{ \theta ^{n}\mid n\in \mathbb{N}\} \cup \{ \sum\nolimits_{n=0}^{N}\theta ^{n}\mid \mathit{\}N\in \mathbb{N}\}$. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such that$\Vert \lambda \alpha \Vert <\frac{1}{% 3}$for all$\alpha \in \{ \sum\nolimits_{n=0}^{N}a_{n}\theta ^{n}\mid$an ∈ {0,1}, N ∈ ℕ}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.

On ℤ-Modules of Algebraic Integers

2009 ◽  
Vol 61 (2) ◽  
pp. 264-281 ◽  
Author(s):  
J. P. Bell ◽  
K. G. Hare
Keyword(s):  
Algebraic Integer ◽  
Full Rank ◽  
Pisot Number ◽  
List Type ◽  
Algebraic Integers ◽  
Pisot Numbers ◽  
Cyclotomic Numbers ◽  
Special Cases ◽  
Fixed Order ◽  
Fixed Positive Integer

Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q. We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d. In this paper we study some properties of ℤ[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results.(1)If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ[qm]´ = disc `ℤ[qn]´.(2)If q and r are algebraic integers of degree d of full rank and ℤ[qn] = ℤ[rn] for infinitely many n, then either q = ωr′ or q = Norm(r)2/dω/r′ , where r ′ is some conjugate of r and ω is some root of unity.(3)Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ[q] = ℤ[r].(4)There are only finitely many Pisot-cyclotomic numbers of any fixed order.

Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point

1997 ◽  
Vol 49 (5) ◽  
pp. 887-915 ◽  
Author(s):  
Peter Borwein ◽  
Christopher Pinner
Keyword(s):  
Algebraic Number ◽  
Real Root ◽  
Small Degree ◽  
Pisot Number ◽  
Roots Of Unity ◽  
Best Approximations ◽  
Rate Of Approximation ◽  
Root Of Unity ◽  
Extremal Polynomials ◽  
Image Position

AbstractFor a fixed algebraic number α we discuss how closely α can be approximated by a root of a {0, +1, -1} polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at α.In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show thatand for a root of unity α thatWe study in detail the case of α = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly.

scholarly journals Salem Numbers and Pisot Numbers via Interlacing

2012 ◽  
Vol 64 (2) ◽  
pp. 345-367 ◽  
Author(s):  
James McKee ◽  
Chris Smyth
Keyword(s):  
Unit Circle ◽  
Rational Functions ◽  
General Construction ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Limit Points ◽  
Zeros And Poles

Abstract We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the “obvious” limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

scholarly journals The Completed Finite Period Map and Galois Theory of Supercongruences

2018 ◽  
Vol 2019 (23) ◽  
pp. 7379-7405
Author(s):  
Julian Rosen
Keyword(s):  
Algebraic Number ◽  
Complex Number ◽  
Galois Theory ◽  
Rational Numbers ◽  
Multiple Zeta Values ◽  
Period Map ◽  
Transcendental Numbers ◽  
Multiple Zeta ◽  
Zeta Values ◽  
Finite Period

Abstract A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.

scholarly journals Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Rafael G. Campos ◽  
Marisol L. Calderón
Keyword(s):  
Closed Form ◽  
Complex Plane ◽  
Complex Number ◽  
Bessel Polynomials ◽  
The Real ◽  
Bessel Polynomial ◽  
Limit Points ◽  
Electrostatic Interpretation ◽  
Approximate Expressions

We find approximate expressionsx̃(k,n,a)andỹ(k,n,a)for the real and imaginary parts of thekth zerozk=xk+iykof the Bessel polynomialyn(x;a). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions ofk,nandais obtained. It is shown that the resulting complex numberx̃(k,n,a)+iỹ(k,n,a)isO(1/n2)-convergent tozkfor fixedk.

scholarly journals Statistical limit point theorems

2000 ◽  
Vol 23 (11) ◽  
pp. 741-752 ◽  
Author(s):  
Jeff Zeager
Keyword(s):  
Complex Number ◽  
Limit Point ◽  
Statistical Convergence ◽  
Nonnegative Matrix ◽  
Bounded Sequence ◽  
Regular Matrix ◽  
Number Sequence ◽  
Statistical Limit ◽  
Limit Points

It is known that given a regular matrixAand a bounded sequencexthere is a subsequence (respectively, rearrangement, stretching)yofxsuch that the set of limit points ofAyincludes the set of limit points ofx. Using the notion of a statistical limit point, we establish statistical convergence analogues to these results by proving that every complex number sequencexhas a subsequence (respectively, rearrangement, stretching)ysuch that every limit point ofxis a statistical limit point ofy. We then extend our results to the more generalA-statistical convergence, in whichAis an arbitrary nonnegative matrix.

Sign in / Sign up

Export Citation Format

Share Document