Bases of \(\varepsilon \)-Canonical Number Systems in Quadratic Number Fields

In this paper, we give an explicit characterization of all bases of \(\varepsilon \) $\epsilon$ -canonical number systems ( \(\varepsilon \) $\epsilon$ -CNS) with finiteness property in quadratic number fields for all values \(\varepsilon \in [0,1)\) $\epsilon \in [0,1)$ . This result is a consequence of the recent result of Jadrijević and Miletić on the characterization of quadratic \(\varepsilon \) $\epsilon$ -CNS polynomials. Our result includes the well-known characterization of all bases of classical CNS ( \(\varepsilon =0\) $\epsilon =0$ ) with finiteness property in quadratic number fields. It also fits into the general framework of generalized number systems (GNS) introduced by A. Pethő and J. Thuswaldner.