Number systems in real quadratic fields (Q2714144)

Let \(I\) be the set of integers in \(\mathbb{Q}(\sqrt D)\) and \(\alpha\in I\), furthermore \(\mathbb A=\{a_0=0,a_1,\dots,a_{|\alpha|-1}\}\) a complete residue system \(\mod \alpha\). We say that \((A,\alpha)\) is a number system if each \(\beta\in I\) has a finite expansion of the form \(\beta=b_0+b_1\alpha+\dots+b_k\alpha^k\) where, \(b_i\in \mathbb A\). The author proves the following theorem. If \(\alpha\) belongs to a real quadratic extension field such that \(|\alpha(1)|>2\), \(|\alpha(2)|>2\) then \((A,\alpha)\) is a number system with a suitable \(A\). This paper is strongly related to a paper of \textit{I. Kátai} [Ann. Univ. Sci. Budap. Rolando Eőtvős, Sect. Comput. 14. 91-103 (1994; Zbl 0817.11046)].