Purely periodic expansions in systems with negative base

Abstract

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 (Fin(β) = Z[β]) is not only sufficient, but also necessary in the case of positive quadratic and cubic bases. We show that Fin(−β) = Z[β] is not necessary in the case of negative bases.

Finite beta-expansions with negative bases

“…d k 0 ω , which can be considered as analogs to simple Parry numbers, do not possess Property (−F). This was conjectured in [19] and supported by the fact that…”

Integers in number systems with positive and negative quadratic Pisot base

“…Relations (16) and (17) describe the possible outcomes of t j ⊕ t k − (t j + t k ) and ⊖t j − (−t j ). By Lemma 5.3, there exists a constant C such that for any pair of factors w, v ∈ L(u −β ) we have |v| 0 − |w| 0 ≤ C. This, together with (16) and (17) shows that t j ⊕ t k − (t j + t k ) and ⊖t j − (−t j ) are bounded independently of j, k, and that when β is a unit, then C = 1 and thus t j ⊕ t k − (t j + t k ), ⊖t j − (−t j ) ∈ 0, ±(∆ 0 − ∆ 1 ) .…”

“…The difference between integers in positive and negative base is illustrated in Figure 1 for the case β = τ = 1 2 (1 + √ 5), the golden ratio. The following proposition appeared as Lemma 5 and 9 in [17].…”