On weak model sets of extremal density

TL;DR

This paper extends the theory of model sets to include weak model sets with boundary of positive measure, establishing their pure point diffraction and dynamical spectrum under extremal density conditions.

Contribution

It introduces a framework for weak model sets with extremal density, proving pure point diffraction and ergodicity, generalizing regular model set results.

Findings

Weak model sets with extremal density have pure point diffraction.

The natural patch frequency measure is ergodic for these sets.

The dynamical spectrum of the hull is pure point under extremal density.

Abstract

The theory of regular model sets is highly developed, but does not cover examples such as the visible lattice points, the k-th power-free integers, or related systems. They belong to the class of weak model sets, where the window may have a boundary of positive measure, or even consists of boundary only. The latter phenomena are related to the topological entropy of the corresponding dynamical system and to various other unusual properties. Under a rather natural extremality assumption on the density of the weak model set we establish its pure point diffraction nature. We derive an explicit formula that can be seen as the generalisation of the case of regular model sets. Furthermore, the corresponding natural patch frequency measure is shown to be ergodic. Since weak model sets of extremal density are generic for this measure, one obtains that the dynamical spectrum of the hull is pure point as well.