Cut-and-project quasicrystals, lattices, and dense forests

TL;DR

This paper investigates the properties of dense forests in Euclidean space, demonstrating that cut-and-project quasicrystals are not dense forests but their unions can be, while unions of lattices often are, with explicit bounds provided.

Contribution

It establishes that cut-and-project quasicrystals are not dense forests, shows finite unions of lattices can be dense forests, and provides explicit constructions and bounds.

Findings

Cut-and-project quasicrystals are not dense forests.

Finite unions of lattices are typically dense forests.

Explicit examples of dense forests with bounds on visibility.

Abstract

Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function. We show that cut-and-project quasicrystals are never dense forests, but their finite unions could be uniformly discrete dense forests. On the other hand, we show that finite unions of lattices typically are dense forests, and give a bound on their visibility function, which is close to optimal. We also construct an explicit finite union of lattices which is a uniformly discrete dense forest with an explicit bound on its visibility.