Pisot Units Research Articles

On Pisot units and the fundamental domain of Galois extensions of [formula omitted]

Let K be a number field that is Galois over Q with unit group of rank d. We obtain an upper bound on the number of facets of the reduction domain (and therefore the fundamental domain) of K of O(((d+1)ρ∞(ΛK)+log⁡(d2))d(d+1)!), where ρ∞(ΛK) is the covering radius of the log-unit lattice in the infinity norm. As a side-result, we also show that the Weil height of the shortest Pisot unit in the number field can be no greater than γ[K:Q]((d+1)ρ∞(ΛK)+dϵ) where γ=1 if K is totally real or 2 otherwise and ϵ>0 is some arbitrarily small constant.

The transcendence of growth constants associated with polynomial recursions

Let [Formula: see text], [Formula: see text], be a polynomial of degree [Formula: see text]. Let [Formula: see text] be a sequence of integers satisfying [Formula: see text] Set [Formula: see text]. Then, under the assumption [Formula: see text], in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either [Formula: see text] is transcendental or [Formula: see text] can be an integer or a quadratic Pisot unit with [Formula: see text] being its conjugate over [Formula: see text]. In this paper, we study the nature of such [Formula: see text] without the assumption that [Formula: see text] is in [Formula: see text], and we prove that either the number [Formula: see text] is transcendental, or [Formula: see text] is a Pisot number with [Formula: see text] being the order of the torsion subgroup of the Galois closure of the number field [Formula: see text]. Other results presented in this paper investigate the solutions of the inequality [Formula: see text] in [Formula: see text], considering whether [Formula: see text] is rational or irrational. Here, [Formula: see text] represents a number field, and [Formula: see text]. The notation [Formula: see text] denotes the distance between [Formula: see text] and its nearest integer in [Formula: see text].

Geometrical representation of subshifts for primitive substitutions

Abstract For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its eigenvalues are roots of unity. Furthermore, we provide an algorithm to compute eigenvalues of the subshift associated with any primitive pseudo-unimodular substitution.

Some class of cubic Pisot numbers with finiteness property

Akiyama classified all cubic Pisot units with finite β-expansions. In this paper, we show a generalization of Akiyama’s theorem by using some reduction theorem. Moreover we give another application of this reduction theorem.

Arithmetics in number systems with cubic base

This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits assuming the finiteness of the expansion of the considered sum or product, especially for the case of cubic Pisot units.

Generalized Rauzy tilings and linear recurrence sequences

Rauzy introduced a fractal set associated with the two-dimensional toric shift by the vector (𝛽−1, 𝛽−2), where 𝛽 is the real root of the equation 𝛽3 = 𝛽2 + 𝛽 + 1 and showed that this fractal is divided into three fractals that are bounded remainder sets with respect to a given toric shift. The introduced set was named as Rauzy fractal. It obtains many applications in the combinatorics of words, geometry, theory of dynamical systems and number theory. Later, an infinite sequence of tilings of 𝑑 − 1-dimensional Rauzy fractals associated with algebraic Pisot units of the degree 𝑑 into fractal sets of 𝑑 types were introduced. Each subsequent tiling is a subdivision of the previous one. These tilings are closely related to some irrational toric shifts and allowed to obtain new examples of bounded remainder sets for these shifts, and also to get some results on self-similarity of shift orbits. In this paper, we continue the study of generalized Rauzy tilings related to Pisot numbers. A new approach to definition of Rauzy fractals and Rauzy tilings based on expansions of natural numbers on linear recurrence sequences is proposed. This allows to improve the results on the connection of Rauzy tilings and bounded remainder sets and to show that the corresponding estimate of the remainder term is independent on the tiling order. The geometrization theorem for linear recurrence sequences is proved. It states that the natural number has a given endpoint of the greedy expansion on the linear recurrence sequence if and only if the corresponding point of the orbit of toric shift belongs to some set, which is the union of the tiles of the Rauzy tiling. Some number-theoretic applications of this result is obtained. In conclusion, some open problems related to generalized Rauzy tilings are formulated.

Lattice Bounded Distance Equivalence for 1D Delone Sets with Finite Local Complexity

Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we characterize, dependingly on digits in the corresponding numeration systems, the spectra which are bounded distance to an average lattice. Our method stems in interpretation of the spectra in the frame of the cut-and-project method. Such structures are coded by an infinite word over a finite alphabet which enables us to exploit combinatorial notions such as balancedness, substitutions and the spectrum of associated incidence matrices.

The Best Approximation of Algebraic Numbers by Multidimensional Continued Fractions

The karyon-module ( $$ \mathcal{K}\mathrm{\mathcal{M}}\hbox{-} $$ ) algorithm for expanding an algebraic number α = (α1, . . . , αd) from ℝd in a multidimensional continued fraction, i.e., in a sequence of rational numbers $$ \frac{P_a}{Q_a}=\left(\frac{P_1^a}{Q^a},\dots, \frac{P_d^a}{Q^a}\right),\kern1em a=1,2,3,\dots, $$ from ℚd with numerators $$ {P}_1^a,\dots, {P}_d^a\in \mathrm{\mathbb{Z}} $$ and a common denominator Qa = 1, 2, 3, . . . is proposed. The $$ \mathcal{K}\mathrm{\mathcal{M}} $$ -algorithm belongs to the class of tuned algorithms and is based on constructing localized Pisot units ζ > 1, for which the moduli of all the conjugates ζ(i) ≠ ζ are contained in the θ- neighborhood of the number ζ−1/d, where the parameter θ may take an arbitrary fixed value. It is proved that given a real algebraic point α of degree deg(α) = d + 1, the $$ \mathcal{K}\mathrm{\mathcal{M}} $$ -algorithm provides its approximation such that $$ \left|\alpha -\frac{P_a}{Q_a}\right|\le \frac{c}{Q_a^{1+\frac{1}{a}-\theta }} $$ for all a ≥ aα,θ, where the constants aα,θ > 0 and c = cα,θ > 0 are independent of a = 1, 2, 3, . . ., and the convergents $$ \frac{P_a}{Q_a} $$ are computed from a certain recurrence relation with constant coefficients, determined by the choice of the localized unit ζ. Bibliography: 19 titles.

Generalized Rauzy tilings and bounded remainder sets

Rauzy introduced a fractal set associted with the toric shift by the vector (β − 1 , β − 2 ), where β is the real root of the equation β 3 = β 2 + β + 1. He show that this fractal can be partitioned into three fractal sets that are bounded remaider sets with respect to the considered toric shift. Later, the introduced set was named as the Rauzy fractal. Further, many generalizations of Rauzy fractal are discovered. There are many applications of the generalized Rauzy fractals to problems in number theory, dynamical systems and combinatorics. Zhuravlev propose an infinite sequence of tilings of the original Rauzy fractal and show that these tilings also consist of bounded remainder sets. In this paper we consider the problem of constructing similar tilings for the generalized Rauzy fractals associated with algebraic Pisot units. We introduce an infinite sequence of tilings of the d −1-dimensional Rauzy fractals associated with the algebraic Pisot units of the degree d into fractal sets of d types. Each subsequent tiling is a subdivision of the previous one. Some results describing the self-similarity properties of the introduced tilings are proved. Also, it is proved that the introduced tilings are so called generalized exchanding tilings with respect to some toric shift. In particular, the action of this shift on the tiling is reduced to exchanging of d central tiles. As a corollary, we obtain that the Rauzy tiling of an arbitrary order consist of bounded remainder sets with respect to the considered toric shift. In addition, some self-similarity property of the orbit of considered toric shift is established.

Multiple tilings associated to d-Bonacci beta-expansions

Let $$\beta \in (1,2)$$ be a Pisot unit and consider the symmetric $$\beta $$ -expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For $$\beta $$ a d-Bonacci number, i.e., Pisot root of $$x^d-x^{d-1}-\dots -x-1$$ we show that the Rauzy fractals form a multiple tiling with covering degree $$d-1$$ .

Pisot unit generators in number fields

Pisot numbers are real algebraic integers bigger than 1, whose other conjugates all have modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called complex Pisot numbers and we solve it in all number fields that admit such a generator, in particular this includes all fields without CM.

Finiteness in real real cubic fields

We study finiteness property in numeration systems with cubic Pisot unit base. A base β > 1 is said to satisfy property (F), if the set Fin (β) of numbers with finite β-expansions forms a ring. We show that in every real cubic field which is not totally real, there exists a cubic Pisot unit satisfying (F). On the other hand, there exist totally real cubic fields without such a unit. In such fields, however, one finds a cubic Pisot unit β > 1 satisfying property (−F), i.e., the set Fin (−β) of finite (−β)-expansions forms a ring.

Pisot Units, Salem Numbers, and Higher Dimensional Projective Manifolds with Primitive Automorphisms of Positive Entropy

We show that, in any dimension greater than one, there are an abelian variety, a smooth rational variety and a Calabi-Yau manifold, with primitive birational automorphisms of first dynamical degree $>1$. We also show that there are smooth complex projective Calabi-Yau manifolds and smooth rational manifolds, of any even dimension, with primitive biregular automorphisms of positive topological entropy.

Spectral properties of cubic complex Pisot units

For a real number $\beta>1$, Erdős, Joo and Komornik study distances between consecutive points in the set $X^m(\beta)=\bigl\{\sum_{j=0}^n a_j \beta^j : n\in\mathbb N,\,a_j\in\{0,1,\dots,m\}\bigr\}$. Pisot numbers play a crucial role for the properties of $X^m(\beta)$. Following the work of Zaimi, who considered $X^m(\gamma)$ with $\gamma\in\mathbb{C}\setminus\mathbb{R}$ and $|\gamma|>1$, we show that for any non-real $\gamma$ and $m < |\gamma|^2-1$, the set $X^m(\gamma)$ is not relatively dense in the complex plane. Then we focus on complex Pisot units with a positive real conjugate $\gamma'$ and $m > |\gamma|^2-1$. If the number $1/\gamma'$ satisfies Property (F), we deduce that $X^m(\gamma)$ is uniformly discrete and relatively dense, i.e., $X^m(\gamma)$ is a Delone set. Moreover, we present an algorithm for determining two parameters of the Delone set $X^m(\gamma)$ which are analogous to minimal and maximal distances in the real case $X^m(\beta)$. For $\gamma$ satisfying $\gamma^3 + \gamma^2 + \gamma - 1 = 0$, explicit formulas for the two parameters are given.

Description of spectra of quadratic Pisot units

The spectrum of a real number β>1 is the set Xm(β) of p(β) where p ranges over all polynomials with coefficients restricted to A={0,1,…,m}. For a quadratic Pisot unit β, we determine the values of all distances between consecutive points and their corresponding frequencies, by recasting the spectra in the frame of the cut-and-project scheme. We also show that shifting the set A of digits so that it contains at least one negative element, or considering negative base −β instead of β, the gap sequence of the modified spectrum is a coding of an exchange of three intervals.

Purely periodic expansions in systems with negative base

We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama’s result for positive Pisot unit base β, we find a sufficient condition so that there exist an interval J containing the origin such that the (−β)-expansion of every rational number from J is purely periodic. We focus on the case of quadratic bases and demonstrate the following difference between the negative and positive bases: It is known that the finiteness property ( $\mathop {\mathrm {Fin}}\nolimits \, (\beta)= {\mathbb {Z}}[\beta]$ ) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that $\mathop {\mathrm {Fin}}\nolimits \, (-\beta)= {\mathbb {Z}}[\beta]$ is not necessary in the case of negative bases.

Periodicity ofβ-expansions for certain Pisot units

Given β > 1, let T β \begin{eqnarray} T_{\beta}:[0,1[ & \rightarrow& [0,1[ \hspace*{0.5 cm} x & \rightarrow & \beta x -\lfloor \beta x \rfloor. \nonumber \end{eqnarray} T β : [ 0 , 1 [ → [ 0 , 1 [ x → βx − ⌊ βx ⌋ . The iteration of this transformation gives rise to the greedy β -expansion. There has been extensive research on the properties of this expansion and its dependence on the parameter β .In [17], K. Schmidt analyzed the set of periodic points of T β , where β is a Pisot number. In an attempt to generalize some of his results, we study, for certain Pisot units, a different expansion that we call linear expansion \begin{eqnarray} x=\sum_{i \geq 0} e_i \beta^{-i},\nonumber \end{eqnarray} x = ∑ i ≥ 0 e i β − i , where each e i can be superior to ⌊ β ⌋, its properties and the relation with Per (β ).

PURELY PERIODIC BETA-EXPANSIONS OVER LAURENT SERIES

The aim of this paper is to characterize the formal power series which have purely periodic β-expansions in Pisot or Salem unit base under some condition. Furthermore, we will prove that if β is a quadratic Pisot unit base, then every rational f in the unit disk has a purely periodic β-expansion and discuss their periods.

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits.

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.