We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots $\beta>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-\beta)$ of finite $(-\beta)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $\beta=\tau=\frac12(1+\sqrt5)$, the golden ratio. For such $\beta$, we determine the exact bound on the number of fractional digits appearing in arithmetical operations. We also show that the set of $(-\tau)$-integers coincides on the positive half-line with the set of $(\tau^2)$-integers.
In practically all fields of applied sciences one meets problems requiring efficient computational methods. An indispensable key for developing such methods is to have fast algorithms for performing arithmetical operations with high precision. The first of the two aspects -speedcan be reached for example using parallelization of algorithms for addition and multiplication. However, it has been shown [15] that this can only be achieved allowing redundancy in number representation. The second aspect -accuracy -calls for special treatment of different classes of irrational numbers by using e.g. exact arithmetics in algebraic number fields. All this motivates the study of non-standard number systems.
Usually, one represents numbers in the standard positional number system with base 10 or base 2, (the so-called decimal or binary representation of numbers). Changing the base for any integer b ≥ 2 does not bring much new. In 1957, Rényi [17] introduced the possibility of representing numbers in a system with non-integer base β > 1. For every non-negative real number x, we have the β-expansion of x of the form
where the digits x i are obtained by the greedy algorithm. In analogy with standard numeration with integer base, we define the set Z β of β-integers, which have vanishing digits at negative powers of β. We also define the set Fin(β) of numbers with finite β-expansion, i.e. numbers whose β-expansion has only finitely many non-zero digits. Many new interesting phenomena appear when considering β-expansions for non-integer base β. For example, whereas in base b ∈ N, every finite string of non-negative integers < b is admissible as the greedy expansion of some x, in base β / ∈ N, this is no longer true. The conditions for admissibility of digit strings for general base β has been given by Parry [16] using the lexicographical ordering of digit strings.
But probably the most remarkable novelty is that Z β is no longer equal to the set Z of rational integers; its elements are not equidistant on the real line and Z β is not a ring (i.e. closed under addition and multiplication), as it is the case for Z. Even more strange, addition of β-integers may result in an infinite β-expansion. Such properties were studied by many authors. Among the most important results on arithmetics with β-expansions is Schmidt’s description of bases β for which rational numbers have periodic β-expansions [18], or the necessary condition on β, so that Fin(β) is a ring, given by Frougny and Solomyak [9]. Others have studied the fractional part appearing in arithmetical operations with β-expansions, see e.g. [5,11,4]. On-line computability of arithmetic operations was studied in [7,10]. Let us note that many questions about arithmetics in the numeration systems with positive real base β remain open.
Recently, Ito and Sadahiro [12] suggested to study positional systems with negative base -β, where β > 1. Here one obtains a representation of every (both positive or negative) real number in the form
Ito and Sadahiro have provided a condition for admissibility of digit strings as (-β)-expansions and shown some properties of the dynamical system connected to (-β)-numeration. Their work on dynamical aspects has been continued in [8]. The authors of [1] define the set Z -β of (-β)integers and focus on its geometrical features. Some arithmetical properties of (-β)-numeration systems are studied in [14]. Among other, the validity of a conjecture of Ito and Sadahiro is established, which states that if β > 1 is the root of x 2 -mx + n, m, n ∈ N, m ≥ n + 2 ≥ 3, then the set of Fin(-β) of finite (-β)-expansions is a ring. One also provides bounds on the length of the fractional part arising by adding and multiplying (-β)-integers, this in case that β is a root of x 2 -mx -1 for m ≥ 2, and that β is a root of x 2 -mx + 1 for m ≥ 3.
In the present paper we complete the arithmetical study for quadratic negative bases started in [14]. In particular, we focus on roots β > 1 of polynomials x 2 -mx -n, m ≥ n ≥ 1, and show that in this case the set Fin(-β) of finite (-β)-expansions is closed under addition, although it is not closed under subtraction. We also provide exact bound on the number of fractional digits appearing in arithmetical operations for the golden ratio τ = 1 2 (1 + √ 5), which is a case missing in the study [14]. We also prove a curious coincidence between (-τ )-integers and (τ 2 )-integers. For that, we need to describe the distances between consecutive (-τ )-integers and morphism under which the infinite word coding their ordering is invariant.
Let β > 1. The Rényi β-expansion of a real number x ∈ [0, 1) can be found as a coding of the orbit of the point x under the transformation T β : [0, 1) → [0, 1), given by the prescription
Directly from the definition of the transformation T β we can derive that the digits
. The number x is thus represented by the infinite word
From the definition of the transformation β we can derive another i
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