Simultaneous visibility in the algebraic lattice

Let $K$ be a number field with ring of integers $\mathcal{O}$. Two lattice points ${\bf x, y}\in \mathcal{O}^m$ with $m\geq 2$ are said to be visible from one another if $\gcd((x_i-y_i),\ldots, (x_m-y_m))=\mathcal{O}$, where $(x_i-y_i)$ is the ideal generated by $x_i-y_i$. Let $S\subset \mathcal{O}^m$ be a finite set. For $K=\mathbb{Q}$, the asymptotic density of the set of lattice points, visible from all points of $S$, was studied by several authors. For general number fields $K$, however, the asymptotic density has been studied only in the special case $S=\{(0,\ldots,0)\}$. Our main result establishes the corresponding density formula for a number field $K$ whose ring of integers $\mathcal{O}$ is a principal ideal domain, for all finite sets $S$ with $|S|\geq 2$.