Rational points in Cantor sets in the complex plane

Let $K$ be an imaginary quadratic field and let ${\mathcal{O}}_K$ be the ring of algebraic integers of $K$. For $\alpha \in {\mathcal{O}}_K$ with $\mathrm{\mid }\alpha \mathrm{\mid }>1$, define $${\mathcal{D}}_{\alpha }=\bigcup _{n=0}^{\mathrm{\infty }}\frac{{\mathcal{O}}_K}{{\alpha }^n}\mathrm{.}$$ For $\beta \in {\mathcal{O}}_K$ with $\mathrm{\mid }\beta \mathrm{\mid }>1$ and a finite subset $A\subset {\mathcal{O}}_K$, define $$S_{\beta ,A}=\{\sum _{k=1}^{\mathrm{\infty }}\frac{a_k}{{\beta }^k}:a_k\in A\mathrm{\forall }k\in \mathbb{N}\}\mathrm{.}$$ Suppose that $\alpha$ and $\beta$ are relatively prime. In this paper, we show that if ${\dim }_{\mathrm{H}}{S}_{\beta ,A}<1$, then the intersection ${\mathcal{D}}_{\alpha }\cap S_{\beta ,A}$ is a finite set. In general, the threshold for the Hausdorff dimension of $S_{\beta ,A}$ is sharp. If we further assume that ${\mathcal{O}}_K$ is a unique factorization domain and that $\bar{\alpha }$ and $\alpha$ are relatively prime, then we establish the finiteness of the intersection under the weaker condition ${\dim }_{\mathrm{H}}{S}_{\beta ,A}<2$. This extends the previously known results on the real line.