Simultaneous visibility in the algebraic lattice

TL;DR

This paper extends the concept of simultaneous visibility among lattice points in algebraic number fields, providing a density formula for cases where the ring of integers is a principal ideal domain and the set of reference points has at least two elements.

Contribution

It generalizes previous results by deriving a density formula for simultaneous visibility in algebraic lattices over number fields with principal ideal rings, for arbitrary finite sets.

Findings

Density formula established for principal ideal domains

Results applicable to all finite sets with at least two points

Extends known results from rational to algebraic number fields

Abstract

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 $|S|\geq 2$.