Asymptotic estimates of large gaps between directions in certain planar quasicrystals

TL;DR

This paper investigates the asymptotic behavior of large gaps between directions in specific planar quasicrystals, extending understanding of their statistical properties and tail distributions.

Contribution

It provides detailed asymptotic estimates for the tail behavior of the gap distribution in certain planar quasicrystals, building on prior invariant process results.

Findings

Derived asymptotic estimates for large gap tail probabilities

Identified specific classes of planar quasicrystals with distinct gap behaviors

Extended statistical understanding of directions in quasicrystals

Abstract

For quasicrystals of cut-and-project type in $\mathbb{R}^d$, it was proved by Marklof and Str\"ombergsson that the limit local statistical properties of the directions to the points in the set are described by certain $\operatorname{SL}_d(\mathbb{R})$-invariant point processes. In the present paper we make a detailed study of the tail asymptotics of the limiting gap statistics of the directions, for certain specific classes of planar quasicrystals.