On the error bounds for visible points in some cut-and-project sets

TL;DR

This paper investigates the distribution of visible points in cut-and-project sets, providing error bounds for their density and showing that such points are not relatively dense in certain cases, advancing understanding of their geometric properties.

Contribution

It establishes error bounds for the density of visible points and proves their non-relative density in specific irreducible cut-and-project sets with star-shaped windows.

Findings

Error bounds for the density of visible points

Visible points are not relatively dense in certain sets

Advances understanding of geometric distribution in cut-and-project sets

Abstract

We study points in cut-and-project sets which are visible from the origin, continuing a direction of inquiry initiated in [6,14] where the asymptotic density of visible points was investigated. We establish an error bound for the density of visible points in certain cases. We also prove that the set of visible points in irreducible cut-and-project sets with star-shaped windows is never relatively dense.