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 $α\in \mathcal{O}_K$ with $|α| > 1$, define \[ \mathcal{D}_α= \bigcup_{n=0}^\infty \frac{\mathcal{O}_K}{α^n}. \] For $β\in \mathcal{O}_K$ with $|β|>1$ and a finite subset $A \subset \mathcal{O}_K$, define \[ S_{β,A} = \bigg\{ \sum_{k=1}^{\infty} \frac{a_k}{β^k}: \; a_k \in A \;\forall k \in \mathbb{N} \bigg\}. \] Suppose that $α$ and $β$ are relatively prime. In this paper, we show that if $\dim_{\mathrm{H}} S_{β,A} < 1$, then the intersection $\mathcal{D}_α\cap S_{β,A}$ is a finite set. In general, the threshold for the Hausdorff dimension of $S_{β,A}$ is sharp. If we further assume that $\mathcal{O}_K$ is a unique factorization domain and that $\overlineα$ and $α$ are relatively prime, then we establish the finiteness of the intersection under the weaker condition $\dim_{\mathrm{H}} S_{β,A} < 2$. This extends the previously known results on the real line.