Abstract. We define the analogue of $q$-additivity for the canonical number systems in the Gaussian ring of integers. We characterize all those functions $f\colon\zz [i]\to\cc $ which are $\theta =-A+i$-additive and completely multiplicative (Theorem 1). For $\theta =-1+i$ we give all functions which are $\theta $- and $\overline{\theta }$-additive (Theorem 2).

DOI: 10.14232/actasm-015-052-9

AMS Subject Classification (1991): 11K65, 11N37, 11N64

Keyword(s): completely additive, completely multiplicative, $q$-additive function, Gaussian integers, canonical number system

Received July 4, 2015, and in revised form June 25, 2016. (Registered under 52/2015.)