Additive functions with respect to expansions over the set of Gaussian integers (Q2759126)

Let \(\mathbb Z[i]\) be the ring of Gaussian integers, \(\theta \in \mathbb Z[i]\) with \(t=|\theta|^2\geq 2\) and \(\mathcal A=\{a_0=0,a_1,\dots ,a_{t-1}\}\subset \mathbb Z[i]\) a complete residue system mod \(\theta\). Let \(\alpha=b_0+b_1\theta+\dots +b_{k-1}\theta^{k-1}+\theta^k\alpha_k\) be the so called correct expansion (which is unique) of \(\alpha \in \mathbb Z[i]\), where \(b_0, b_1,\dots,b_{k-1}\in \mathcal A\) are the digits of the expansion. The authors define the concepts of additive and multiplicative functions \(f:\mathbb Z[i] \to \mathbb R\) with respect to the expansion of above. They investigate the existence of the main value of such multiplicative functions \(f\) with \(|f(\alpha)|=1\), \(\alpha \in \mathbb Z[i]\) and the limit distribution of additive functions (in the above sense). It is shown that the analogue of Delange's theorem concerning \(q\)-multiplicative functions [see \textit{H. Delange}, Acta Arith. 21, 285-298 (1972; Zbl 0219.10062)] remains valid, and a result for the local distribution of the sum-of-digits function is proved.