On the Finiteness property of negative cubic Pisot bases

Abstract

We study arithmetical aspects of Ito-Sadahiro number systems with negative base. We show that the bases $-\beta <-1$, where $\beta$ is zero of $x^3-m{x}^2-mx-m,m\in \mathbb{N},$ possess the so-called finiteness property. For the Tribonacci base $-\gamma ,$ zero of $x^3-x^2-x-1$, we present an effective algorithm for addition and subtraction. In particular, we present a finite state transducer performing these operations. As a consequence of the structure of the transducer, we determine the maximal number of fractional digits arising from addition or subtraction of two $(-\gamma )$-integers.

13pp