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 $|\alpha | \gt 1$, define \[ \mathcal {D}_\alpha = \bigcup_{n=0}^\infty \frac{\mathcal {O}_K}{\alpha ^n}. \] For $\beta \in \mathcal {O}_K$ with $|\beta | \gt 1$ and a finite subset $A \subset \mathcal {O}_K$, define \[ S_{\beta ,A} = \biggl\{ \sum _{k=1}^{\infty } \frac{a_k}{\beta ^k}: a_k \in A \ \forall k \in \mathbb {N} \biggr\}. \] Suppose that $\alpha $ and $\beta $ are relatively prime. In this paper, we show that if $\dim_{\mathrm {H}} S_{\beta ,A} \lt 1$, 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 $\dim_{\mathrm {H}} S_{\beta ,A} \lt 2$. This extends the previously known results on the real line.
