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Ito-Sadahiro Numbers Vs. Parry Numbers

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Ito-Sadahiro Numbers Vs. Parry Numbers Acta Polytechnica Vol. 51 No. 4/2011 Ito-Sadahiro numbers vs. Parry numbers Z. Mas´akov´a, E. Pelantov´a Abstract We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which β happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of − is eventually periodic. We call β +1 such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in thesamecontextfortheR´enyi positive base numeration system. Keywords: numeration systems, negative base, Pisot number, Parry number. ∈AN 1 Introduction According to Parry, the string x1x2x3 ... rep- resents the β-expansion of a number x ∈ [0, 1) if and The expansion of a real number in the positional only if number system with base β>1, as defined by R´enyi [12] is closely related to the transformation ∗ xixi+1xi+2 ...≺ d (1) (2) T :[0, 1) → [0, 1), given by the prescription T (x):= β βx −#βx$.Everyx ∈ [0, 1) is a sum of the infinite for every i =1, 2, 3,... series D { | ∞ Condition (2) ensures that the set β = dβ(x) xi − ∈ } D x = , where x = #βT i 1(x)$ (1) x [0, 1) is shift invariant, and so the closure of β βi i N i=1 in A , denoted by Sβ, is a subshift of the full shift N for i =1, 2, 3,... A . The notion of β-expansion can naturally be ex- Directly from the definition of the transformation T tended to all non-negative real numbers: The expres- we can derive that the ‘digits’ x take values in the set i sion of a positive real number y in the form {0, 1, 2,...,%β&−1} for i =1, 2, 3,.... The expres- sion of x in the form (1) is called the β-expansion of k k−1 k−2 x.Thenumberx is thus represented by the infinite y = ykβ + yk−1β + yk−2β + ... , (3) N ∈ Z ∈D word dβ(x)=x1x2x3 ... ∈A over the alphabet where k and ykyk−1yk−2 ... β, A = {0, 1, 2,...,%β&−1}. From the definition of the transformation T we is called the β-expansion of y. can derive another important property, namely that Real numbers y having in the β-expansion of |y| the ordering on real numbers is carried over to the vanishing digits yi for all i<0 are usually called β- ordering of β-expansions. In particular, we have for integers, and the set of β-integers is denoted by Zβ. x, y ∈ [0, 1) that The notion of β-integers was first considered in [3] as an aperiodic structure modeling non-crystallographic x ≤ y ⇐⇒ d (x) ) d (y) , β β materials with long range order, called quasicrystals. N Numbers y with finitely many non-zero digits in the where ) is the lexicographical order on A ,(order- β-expansion of |y| form the set denoted by Fin(β). ing on the alphabet A is usual, 0 < 1 < 2 < ... < The choice of the base β>1 strongly influences %β&−1). the properties of β-expansions. It turns out that an In [11], Parry has provided a criterion which de- N important role among bases is played by such num- cides whether an infinite word in A is or not a β- ∗ bers β for which dβ(1) is eventually periodic. Parry expansion of some real number x. The criterion is himself called these bases beta-numbers; now these formulated using the so-called infinite expansion of ∗ numbers are commonly called Parry numbers. We 1, denoted by dβ(1), defined as a limit in the space can demonstrate the exceptional properties of Parry N A equipped with the product topology, by numbers on two facts: • The subshift S is sofic if and only if β is a Parry ∗ − β dβ(1) := lim dβ(1 ε) . ε→0+ number [6]. 59 Acta Polytechnica Vol. 51 No. 4/2011 • Distances between consecutive β-integers take defines a sofic system if and only if d−β(lβ)iseven- finitely many values if and only if β is a Parry tually periodic. number [15]. By analogy with the definition of Parry numbers, Recently, Ito and Sadahiro [5] suggested a study we suggest that numbers β>1 such that d−β(lβ)is of positional numeration systems with a negative eventually periodic be called Ito-Sadahiro numbers. base −β,whereβ>1. The representation of real The relation of the set of Ito-Sadahiro numbers and numbers in such a system is defined using the trans- the set of Parry numbers is not obvious. Bassino [2] β has shown that quadratic numbers, as well as cubic formation T :[lβ,rβ ) → [lβ,rβ), where lβ = − , β +1 numbers which are not totally real, are Parry if and 1 only if they are Pisot. For the same class of num- rβ =1+lβ = , 1+β bers, we prove in [10] that β is Ito-Sadahiro if and only if it is Pisot. This means that notions of Parry − −#− − $ T (x):= βx βx lβ . (4) numbers and Ito-Sadahiro numbers on the mentioned type of irrationals do not differ. This would support Every real x ∈ I := [l ,r ) can be written as β β β the hypothesis stated in the first version of this pa- ∞ per, namely that the set of Parry numbers and the x x = i , (5) set of Ito-Sadahiro numbers coincide. However, dur- (−β)i i=1 ing the refereeing process Liao and Steiner [9] found an example of a Parry number which is not an Ito- #− i−1 − $ where xi = βT (x) lβ for i =1, 2, 3,... Sadahiro number, and vice-versa. The above expression is called the (−β)- The main results of this paper are formulated as expansion of x. It can also be written as the infi- Theorems 4 and 7. Theorem 4 gives a bound on the nite word d−β(x)=x1x2x3 ... We can easily show modulus of conjugates of Ito-Sadahiro numbers; The- ≥ from (4) that the digits xi, i 1, take values in the orem 7 shows that periodicity of (−β)-expansion of set A = {0, 1, 2,...,#β$}. In this case, the order- all numbers in the field Q(β)requiresβ to be a Pisot ing on the set of infinite words over the alphabet A or Salem number. Statements which we prove, as well which would correspond to the ordering of real num- as results of other authors that we recall, demonstrate bers is the so-called alternate ordering: We say that similarities between the behaviour of β-expansions ≺ x1x2x3 ... alt y1y2y3 ... if for the minimal index j and (−β)-expansions. We mention also phenomena − j − j such that xj = yj it holds that xj ( 1) <yj( 1) . in which the two essentially differ. In this notation, we can write for arbitrary x, y ∈ Iβ that ≤ ⇐⇒ ) 2 Preliminaries x y d−β(x) alt d−β(y) . Letusfirstrecallsomenumber theoretical notions. A In their paper, Ito and Sadahiro have provided a N complex number β is called an algebraic number, if it A n n−1 criterion to decide whether an infinite word be- is a root of a monic polynomial x +a − x +...+ − n 1 longs to the set of ( β)-expansions, i.e. to the set a x + a , with rational coefficients a ,...,a − ∈ Q. D { | ∈ } 1 0 0 n 1 −β = d−β(x) x Iβ . This time, the criterion is A monic polynomial with rational coefficients and given in terms of two infinite words, namely root β of the minimal degree among all polynomials ∗ − with the same properties is called the minimal poly- d−β(lβ)andd−β(rβ) := lim d−β(rβ ε) . ε→0+ nomial of β, and its degree is called the degree of β. The roots of the minimal polynomial are algebraic These two infinite words have a close relation: If conjugates. d−β(lβ) is purely periodic with odd period length, i.e. If the minimal polynomial of β has integer coeffi- ω ∗ d−β(lβ)=(d1d2 ...d2k+1) ,thenwehaved−β(rβ )= cients, β is called an algebraic integer. An algebraic ω ω 0d1d2 ...(d2k+1 − 1) . (As usual, the notation w integer β>1 is called a Perron number, if all its con- stands for infinite repetition of the string w.) In all jugates are in modulus strictly smaller than β.An ∗ other cases we have d−β(rβ )=0d−β(lβ). algebraic integer β>1 is called a Pisot number, if all Ito and Sadahiro have shown that an infinite its conjugates are in modulus strictly smaller than 1. word x1x2x3 ...represents a (−β)-expansion of some An algebraic integer β>1 is called a Salem number, x ∈ [lβ,rβ) if and only if for every i ≥ 1 it holds that if all its conjugates are in modulus smaller than or equal to 1 and β is not a Pisot number. ) ≺ ∗ d−β(lβ) alt xixi+1xi+2 ... alt d−β(rβ) . (6) If β is an algebraic number of degree n, then the minimal subfield of the field of complex numbers con- D The above condition ensures that the set −β of in- taining β is denoted by Q(β) and is of the form finite words representing (−β)-expansions is shift in- n−1 variant. In [5] it is shown that the closure of D−β Q(β)={c0 + c1β + ...+ cn−1β | ci ∈ Q} .
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