@article{c57d08d880ec4e5e9ac9ecf003a4c64f,
title = "Cut-and-project quasicrystals, lattices and dense forests",
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.",
author = "Faustin Adiceam and Yaar Solomon and Barak Weiss",
note = "Funding Information: The authors are grateful to the anonymous referee for helpful suggestions. They also thank the referee for producing Figure 2 and agreeing to reproduce it in this paper. The authors gratefully acknowledge the support of grants EP/T021225, BSF 2016256 and ISF 2095/15. The first author wishes to thank Federico Ardila for a talk given at the University of Waterloo in 2018 which turned out to be most illuminating to solve some of the questions raised in this paper. Publisher Copyright: {\textcopyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",
year = "2022",
month = mar,
day = "1",
doi = "10.1112/jlms.12534",
language = "English",
volume = "105",
pages = "1167--1199",
journal = "Journal of the London Mathematical Society",
issn = "0024-6107",
publisher = "John Wiley and Sons Ltd",
number = "2",
}