On symmetric radix representation of Gaussian integers (Q1825898)

The main result reads as follows: Let \(m\in\mathbb Z[i]\) be a fixed Gaussian integer satisfying \(N(m)\geq 5\). If \(z\) is a Gaussian integer such that \(-\tfrac12<\text{Re}(z/m^ n)\), \(\text{Im}(z/m^ n)\leq\tfrac12\) then \(z\) has a unique representation of the form \(z=z_ 0+z_ 1m+...+z_{n-1}m^{n-1}+qm^ n\) with \(z_ j\in\mathbb Z[i]\), \(-\tfrac12<\text{Re}(z_ i/m)\), \(\text{Im}(z_ i/m)\leq\tfrac12\), \(1\leq j<n\) and \(q\in \{0,\pm 1,\pm i,\pm 1\pm i\}\). This result shows that this representation gives a ``complex number system'' in the sense of \textit{I. Kátai} and \textit{J. Szabó} [Acta Sci. Math. 37, 255--260 (1975; Zbl 0297.12003)].