Gap asymptotics of the directions in an Ammann-Beenker-like quasicrystal

Abstract

It is known that the limiting gap distribution of the directions to visible points in planar quasicrystals of cut-and-project type exists as a continuous function $F(s)$. In this article we study the asymptotic behaviour of said limiting gap distribution in the particular case of an Ammann--Beenker-like quasicrystal $\mathcal{P}$; more precisely we show that in this case $F(s)=C_{\mathcal{P}}{s}^{-2}+\mathcal{O}(s^{-17/8})$ as $s\to \infty$ with an explicit constant $C_{\mathcal{P}}>0$.