Rational points in Cantor sets in the complex plane

Abstract

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 $|\alpha| > 1$, define \[ \mathcal{D}_\alpha = \bigcup_{n=0}^\infty \frac{\mathcal{O}_K}{\alpha^n}. \] For $\beta \in \mathcal{O}_K$ with $|\beta|>1$ and a finite subset $A \subset \mathcal{O}_K$, define \[ S_{\beta,A} = \bigg\{ \sum_{k=1}^{\infty} \frac{a_k}{\beta^k}: \; a_k \in A \;\forall k \in \mathbb{N} \bigg\}. \] 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 $\overline{\alpha}$ 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.