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

TL;DR

This paper investigates the asymptotic behavior of the gap distribution of directions in an Ammann-Beenker-like quasicrystal, revealing a specific power-law decay with an explicit constant and error term.

Contribution

It provides the first detailed asymptotic analysis of the gap distribution for this particular quasicrystal type, including explicit constants and error estimates.

Findings

The gap distribution decays as s^{-2} for large s.

An explicit constant C_P is identified for the Ammann-Beenker-like quasicrystal.

The asymptotic error term is of order s^{-17/8}.

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 $C_{\mathcal{P}}>0$.