Gaussian rational numbers in Cantor sets in the complex plane

TL;DR

This paper investigates the structure of certain Cantor-like sets in the complex plane generated by Gaussian rationals, showing finiteness of Gaussian rationals with restricted prime factors when the set's Hausdorff dimension is below one.

Contribution

It establishes a finiteness result for Gaussian rationals within these sets under a Hausdorff dimension constraint, linking geometric measure theory with algebraic properties.

Findings

Finiteness of Gaussian rationals with restricted prime factors in the set

Connection between Hausdorff dimension and algebraic structure

Extension of classical results to complex Gaussian integers

Abstract

Given $\beta\in\mathbb{Z}[i]$ with $|\beta|>1$ and a finite set $D\subset\mathbb{Q}(i)$, let \[K_{\beta, D}=\left\{\sum_{j=1}^{\infty}\frac{d_j}{\beta^j}: d_j\in D, \forall j\geq 1\right\}.\] Let $\mathcal{S}$ be a finite set of non-associate prime elements in $\mathbb{Z}[i]$ not dividing $\beta$. We prove that if the Hausdorff dimension of $K_{\beta,D}$ is less than $1$, then there are only finitely many Gaussian rational numbers in $K_{\beta,D}$ whose denominators have all their prime factors in $\mathcal{S}$.