Visibility and directions in quasicrystals

Abstract

It is well known that a positive proportion of all points in a $d$-dimensional lattice is visible from the origin, and that these visible lattice points have constant density in $\mathbb{R}^d$. In the present paper we prove an analogous result for a large class of quasicrystals, including the vertex set of a Penrose tiling. We furthermore establish that the statistical properties of the directions of visible points are described by certain $\operatorname{SL}(d,\mathbb{R})$-invariant point processes. Our results imply in particular existence and continuity of the gap distribution for directions in certain two-dimensional cut-and-project sets. This answers some of the questions raised by Baake et al. in [arXiv:1402.2818].