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 $\hat{I}\pm \in {\mathcal{O}}_K$ with $|\hat{I}\pm |>1$, define \[ \mathcal{D}_α= \bigcup_{n=0}^\infty \frac{\mathcal{O}_K}{α^n}. \] For ${\hat{I}}^2\in {\mathcal{O}}_K$ with $|{\hat{I}}^2|>1$ 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 $\hat{I}\pm$ and ${\hat{I}}^2$ are relatively prime. In this paper, we show that if ${\dim }_{\mathrm{H}}{S}_{{\hat{I}}^2,A}<1$, then the intersection ${\mathcal{D}}_{\hat{I}}\pm \cap S_{{\hat{I}}^2,A}$ is a finite set. In general, the threshold for the Hausdorff dimension of $S_{{\hat{I}}^2,A}$ is sharp. If we further assume that ${\mathcal{O}}_K$ is a unique factorization domain and that $\overline{\hat{I}}\pm$ and $\hat{I}\pm$ are relatively prime, then we establish the finiteness of the intersection under the weaker condition ${\dim }_{\mathrm{H}}{S}_{{\hat{I}}^2,A}<2$. This extends the previously known results on the real line.

13 pages