Let $\beta$ be a Gaussian integer whose norm $N$ is at least three, and let $u_1,\ldots,u_N$ be distinct Gaussian integers at least one of which is not divisible by $\beta$. Consider the set $R$ of all Gaussian integers which can be expressed in the form $\sum_{k=0}^n \beta^k u_{i_k}$ for some integer $n$ and choice of indices $i_0,\ldots,i_n$. What conditions on $u_1,\ldots,u_N$ guarantee that there exists an element of $R$ which has at least two such representations?
A necessary condition for there to be such an element is that the numbers $u_i$ should not come from $N$ distinct residue classes. (This is just a simple calculation: compare two representations of the same number, take the difference, divide by the largest possible factor of $\beta$ and note that we have an equation $u_i-u_j=\beta z$ for some Gaussian integer $z$.) Is this condition also sufficient? If not, what is a sufficient condition which holds for a reasonably abundant set of $(u_1,\ldots,u_N)$?
The literature on this precise question is apparently relatively small, but there is a more substantial literature on uniqueness of radix representations $\sum_{k=0}^m d_k A^{-k}$ where $d_k \in \mathbb{Z}^n$ and $A$ is an integer matrix with all eigenvalues larger than $1$ in absolute value. (The case where $A$ is the $2\times 2$ matrix representing multiplication by a Gaussian integer presents the original question as a special case of this framework.) This question arises in the problem of deciding whether a self-affine tile has positive Lebesgue measure. Theorem 4.1 of Lagarias and Wang's 1996 article "Integral self-affine tiles in $\mathbb{R}^n$: standard and non-standard digit sets" implies that when the norm of the Gaussian integer is a prime number, the above condition is both necessary and sufficient, but suggests that it may fail to be sufficient in general.