Suchergebnisse

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 happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of −β/(β + 1) is eventually periodic. We call such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in the same context for the Rényi positive base numeration system.

We study the selfmatching properties of Beatty sequences, in particular of the graph of the function ⌊ jβ ⌋ against j for every quadratic unit βϵ (0,1). We show that translation in the argument by an element Gi of a generalized Fibonacci sequence almost always causes the translation of the value of the function by Gi=1. More precisely, for fixed i ϵ ℕ, we have ⌊β(j+Gi)⌋ = ⌊βj⌋ + Gi=1, where j ϵ Ui. We determine the set Ui of mismatches and show that it has a low frequency, namely βi.