The sum of digits of squares in Z[i]

We consider the complex sum-of-digits function sq for squares with respect to special bases q of a canonical number system in the Gaussian integers Z[i]. In particular, we show that the sequence (αsq(z2))z∈Z[i] is uniformly distributed modulo 1 if and only if α is irrational. Furthermore we introduce special sets of Gaussian integers (related to Følner sequences) for which we can determine the order of magnitude of the number of integers z for which sq(z2) lies in a fixed residue class mod m. This extends a recent result of Mauduit and Rivat to Z[i]. We also improve an estimate of Gittenberger and Thuswaldner in order to show a local limit theorem for the sum-of-digits function of squares. We can provide asymptotic expansions for where (DN)N∈N is a sequence of convex sets.