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

TL;DR

This paper calculates the density of visible points in the Ammann-Beenker point set, extending known lattice results to a quasicrystal structure using Dedekind's zeta function over a quadratic field.

Contribution

It provides a formula for the density of visible points in the Ammann-Beenker set, linking it to Dedekind's zeta function, a novel extension of lattice point density results.

Findings

Density of visible points in Ammann-Beenker set is 2(√2-1)/ζ_K(2)

Connects quasicrystal point set density to algebraic number theory

Extends lattice point density results to a non-periodic structure

Abstract

The relative density of visible points of the integer lattice $\mathbb{Z}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, 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 $K=\mathbb{Q}(\sqrt{2})$.