On weak model sets of extremal density

Author
Baake, Michael
Huck, Christian
Strungaru, Nicolae
Faculty Advisor
Date
2016
Keywords
weak model sets , visible lattice points , positive measure , density
Abstract (summary)
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.
Publication Information
Baake, M., Huck, C., & Strungaru, N. (2016). On weak model sets of extremal density. arXiv:1512.07129v2. https://arxiv.org/abs/1512.07129
DOI
Notes
Item Type
Report
Language
English
Rights
All Rights Reserved