Gaussian rational numbers in Cantor sets in the complex plane

Abstract

Given $\beta\in\mathbb{Z}[i]$ with $|\beta|>1$ and a finite set $D\subset\mathbb{Q}(i)$, let \[K_{\beta, D}=\left\{\sum_{j=1}^{\infty}\frac{d_j}{\beta^j}: d_j\in D, \forall j\geq 1\right\}.\] 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 $\mathcal{S}$.