The density of visible points in the Ammann-Beenker point set

Abstract

The relative density of visible points of the integer lattice ${\mathbb{Z}}^d$ is known to be $1/\zeta (d)$ for $d\ge 2$, where $\zeta$ is Riemann's zeta function. In this paper we prove that the relative density of visible points in the Ammann-Beenker point set is given by $2(\sqrt{2}-1)/{\zeta }_K(2)$, where ${\zeta }_K$ is Dedekind's zeta function over $K=\mathbb{Q}(\sqrt{2})$.