Gaussian rational numbers in Cantor sets in the complex plane

Given $\beta \in \mathbb{Z}[i]$ with $\mathrm{\mid }\beta \mathrm{\mid }>1$ and a finite set $D\subset \mathbb{Q}(i)$, let $$K_{\beta ,D}=\{\sum _{j=1}^{\mathrm{\infty }}\frac{d_j}{{\beta }^j}:d_j\in D,\mathrm{\forall }j\ge 1\}\mathrm{.}$$ Let $\mathcal{S}$ be a finite set of non-associate prime elements in $\mathbb{Z}[i]$ not dividing $\beta$. We prove that if the Hausdorff dimension of $K_{\beta ,D}$ is less than $1$, then there are only finitely many Gaussian rational numbers in $K_{\beta ,D}$ whose denominators have all their prime factors in $\mathcal{S}$.