Pisot Numbers Research Articles

ON THE SPECTRA OF 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))}.

Norms extremal with respect to the Mahler measure

In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmerʼs conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures.

Homological Pisot substitutions and exact regularity

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and the first rational Čech cohomology is d-dimensional. We construct examples of such “homological Pisot” substitutions whose tiling flows do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.

Negative bases and automata

Automata, Logic and Semantics We study expansions in non-integer negative base -beta introduced by Ito and Sadahiro. Using countable automata associated with (-beta)-expansions, we characterize the case where the (-beta)-shift is a system of finite type. We prove that, if beta is a Pisot number, then the (-beta)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer. We then give an on-line algorithm for the conversion from positive base beta to negative base -beta. When beta is a Pisot number, the conversion can be realized by a finite on-line transducer.

Pisot-Fibonacci q-coherent states

A family of q-coherent states is constructed allowing us to obtain a new quantized version of the harmonic oscillator. These q-states are normalized and form an overcomplete set resolving the unity with respect to the appropriate Jackson measure if 0 < q < 1. We only consider here those values of q such that q−1 is a Pisot number. In this case the q-deformed integers ([n]q) form Fibonacci-like sequences of integers. We study the main physical characteristics of the corresponding quantum oscillator: localization in the configuration and in the phase spaces, probability distributions and related statistical features and semi-classical phase space trajectories whose periodicity is related with the fact that q is an algebraic number.

Arithmetics in number systems with a negative base

We study the numeration system with a negative base, introduced by Ito and Sadahiro. We focus on arithmetic operations in the sets Fin ( − β ) and Z − β of numbers having finite resp. integer ( − β ) -expansions. We show that Fin ( − β ) is trivial if β is smaller than the golden ratio 1 2 ( 1 + 5 ) . For β ≥ 1 2 ( 1 + 5 ) we prove that Fin ( − β ) is a ring, only if β is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that Fin ( − β ) is a ring if β is a quadratic Pisot number with positive conjugate. For quadratic Pisot units, we determine the number of fractional digits that may appear when adding or multiplying two ( − β ) -integers.

On the location of roots of non-reciprocal integer polynomials

Let P be a polynomial of degree d with integer coefficients such that P(0) ≠ 0. Assuming that P has no reciprocal factors we obtain a lower bound on the modulus of the smallest root of P in terms of its degree d, its Mahler measure M(P) and the number of roots of P lying outside the unit circle, say, k. We derive from this that all d roots of P must lie in the annulus R 0 }\,(12M(\alpha)^2 \log M(\alpha))^{-d}}$$. Some lower bounds on the moduli of the conjugates of a Pisot number are also given. In particular, it is shown that if α is a cubic Pisot number, then the disc |z| ≤ α −1 + 0.1999α −2 contains no conjugates of α. Here the constant 0.1999 cannot be replaced by the constant 0.2. We also show that if α is a Pisot number of degree at least 4 and α′ is its conjugate, then |α α′ − 1| > (19α 2)−1.

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter r which varies in the d-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings.In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters r these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials.We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter r of the shift radix system, these tiles provide multiple tilings and even tilings of the d-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine).

A lower bound for Garsia’s entropy for certain Bernoulli convolutions

AbstractLet β∈(1,2) be a Pisot number and let Hβ denote Garsia’s entropy for the Bernoulli convolution associated with β. Garsia, in 1963, showed that Hβ&lt;1 for any Pisot β. For the Pisot numbers which satisfy xm=xm−1+xm−2+⋯+x+1 (with m≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier for m=2 and later by Grabner, Kirschenhofer and Tichy for m≥3, and it proves to be close to 1. No other numerical values for Hβ are known. In the present paper we show that Hβ&gt;0.81 for all Pisot β, and improve this lower bound for certain ranges of β. Our method is computational in nature.

Rational numbers with purely periodic β -expansion

We study real numbers β with the curious property that the β-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let γ(β) denote the supremum of the real numbers c in (0, 1) such that all positive rational numbers less than c have a purely periodic β-expansion. We prove that γ(β) is irrational for a class of cubic Pisot units that contains the smallest Pisot number η. This result is motivated by the observation of Akiyama and Scheicher that γ(η) = 0.666 666 666 086 … is surprisingly close to 2/3.

Pisot Numbers from {0, 1}-Polynomials

AbstractA Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

Growth rate for beta-expansions

Let β > 1 and let m > β be an integer. Each \({x\in I_\beta:=[0,\frac{m-1}{\beta-1}]}\) can be represented in the form$$x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},$$ where \({\epsilon_k\in\{0,1,\ldots,m-1\}}\) for all k (a β-expansion of x). It is known that a.e. \({x\in I_\beta}\) has a continuum of distinct β-expansions. In this paper we prove that if β is a Pisot number, then for a.e. x this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by β. When \({\beta < \frac{1+\sqrt5}2}\), we show that the set of β-expansions grows exponentially for every internal x.

Salem numbers defined by Coxeter transformation

A real algebraic integer α>1 is called a Salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1. It is known [J.A. de la Peña, Coxeter transformations and the representation theory of algebras, in: V. Dlab et al. (Eds.), Finite Dimensional Algebras and Related Topics, Proceedings of the NATO Advanced Research Workshop on Representations of Algebras and Related Topics, Ottawa, Canada, Kluwer, August 10–18, 1992, pp. 223–253; J.F. McKee, P. Rowlinson, C.J. Smyth, Salem numbers and Pisot numbers from stars, Number theory in progress. in: K. Győry et al. (Eds.), Proc. Int. Conf. Banach Int. Math. Center, Diophantine problems and polynomials, vol. 1, de Gruyter, Berlin, 1999, pp. 309–319; P. Lakatos, On Coxeter polynomials of wild stars, Linear Algebra Appl. 293 (1999) 159–170] that the spectral radii of Coxeter transformation defined by stars, which are neither of Dynkin nor of extended Dynkin type, are Salem numbers. We prove that the spectral radii of the Coxeter transformation of generalized stars are also Salem numbers. A generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3.

On ℤ-Modules of Algebraic Integers

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.

On numbers having finite beta-expansions

AbstractLet β be a real number greater than one, and let ℤβ be the set of real numbers which have a zero fractional part when expanded in base β. We prove that β is a Pisot number when the set ℕβ−ℕβ−ℕβ is discrete, where ℕβ=ℤβ∩[0,∞[. We also give partial answers to some related open problems, and in particular, we show that β is a Pisot number when a sum ℤβ+⋯+ℤβ is a Meyer set.

Continuity properties and infinite divisibility of stationary distributions of some generalized Ornstein–Uhlenbeck processes

Properties of the law μ of the integral ∫0∞c−Nt− dYt are studied, where c>1 and {(Nt, Yt), t≥0} is a bivariate Lévy process such that {Nt} and {Yt} are Poisson processes with parameters a and b, respectively. This is the stationary distribution of some generalized Ornstein–Uhlenbeck process. The law μ is parametrized by c, q and r, where p=1−q−r, q, and r are the normalized Lévy measure of {(Nt, Yt)} at the points (1, 0), (0, 1) and (1, 1), respectively. It is shown that, under the condition that p>0 and q>0, μc, q, r is infinitely divisible if and only if r≤pq. The infinite divisibility of the symmetrization of μ is also characterized. The law μ is either continuous-singular or absolutely continuous, unless r=1. It is shown that if c is in the set of Pisot–Vijayaraghavan numbers, which includes all integers bigger than 1, then μ is continuous-singular under the condition q>0. On the other hand, for Lebesgue almost every c>1, there are positive constants C1 and C2 such that μ is absolutely continuous whenever q≥C1p≥C2r. For any c>1 there is a positive constant C3 such that μ is continuous-singular whenever q>0 and max {q, r}≤C3p. Here, if {Nt} and {Yt} are independent, then r=0 and q=b/(a+b).

Une remarque sur le spectre des nombres de Pisot

Let θ be a Pisot number less than 2, m a positive rational integer, and A m the set of the polynomials with coefficients in { 0 , 1 , … , m } evaluated at θ. We give a lower bound for the greatest limit point of common differences of consecutive elements of A m . To cite this article: T. Zaïmi, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Scaling dynamics of a cubic interval-exchange transformation

We study the dynamics of renormalization of a specific interval-exchange transformation which features exact scaling (the cubic Arnoux–Yoccoz model). Using a symbolic space that describes both dynamics and scaling, we characterize the periodic points of the scaling map in terms of generalized decimal expansions, where the base is the reciprocal of a Pisot number and the digits are algebraic integers. The set of periodic points has a rich arithmetic and geometric structure: we establish rigorously some facts, and use extensive numerical experimentation to formulate a conjecture.

Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

This paper studies tilings and representation sapces related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). The obtained results are applied to study the set of rational numbers having a purely periodic β-expansion. We indeed make use of the connection between pure periodicity and a compact self-similar representation of numbers having no fractional part in their β-expansion, called central tile: for elements x of the ring \({\mathbb {Z}[1/\beta]}\) , so-called x-tiles are introduced, so that the central tile is a finite union of x-tiles up to translation. These x-tiles provide a covering (and even in some cases a tiling) of the space we are working in. This space, called complete representation space, is based on Archimedean as well as on the non-Archimedean completions of the number field \({{\mathbb Q} (\beta)}\) corresponding to the prime divisors of the norm of β. This representation space has numerous potential implications. We focus here on the gamma function γ(β) defined as the supremum of the set of elements v in [0, 1] such that every positive rational number p/q, with p/q ≤ v and q coprime with the norm of β, has a purely periodic β-expansion. The key point relies on the description of the boundary of the tiles in terms of paths on a graph called “boundary graph”. The paper ends with explicit quadratic examples, showing that the general behaviour of γ(β) is slightly more complicated than in the unit case.

Constructions of Pisot and Salem numbers with flat palindromes

This paper introduces explicit conditions for some natural family of polynomials to define Pisot or Salem numbers, and reviews related topics as well as their references.