Spectrum of weak model sets with Borel windows

TL;DR

This paper extends the understanding of the spectral and measure-theoretic properties of weak model sets with Borel windows, showing that key results hold beyond compact windows and discussing implications for diffraction and generic examples.

Contribution

It generalizes previous results on weak model sets by removing the compactness assumption on the window, using Moody's work on uniform distribution, and explores diffraction and generic examples.

Findings

Extended hull is a measure-theoretic factor of a group rotation.

Results hold for arbitrary measurable, relatively compact windows.

Implications for diffraction and new generic examples discussed.

Abstract

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article "Periods and factors of weak model sets" we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody's work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.