Gaussian rational numbers in Cantor sets in the complex plane

Abstract

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