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

TL;DR

The paper analyzes visible points in cut-and-project sets, establishing both density formulas and rate-of-convergence results. It proves that for irreducible CPS with star-shaped windows and almost every lattice in GL(n,R), the density of visible points equals the full density scaled by the reciprocal zeta factor, with explicit power-saving error bounds; sharper rates are obtained for Hammarhjelm examples. A random CPS framework using RMS measures shows that, for almost every CPS, visibility coincides with the lattice-visibility construction and yields a universal density formula with tractable counting errors. In the real-quadratic setting under the Hammarhjelm condition, the authors derive explicit density expressions in terms of the fundamental unit and Dedekind zeta, and provide effective inclusion–exclusion methods to obtain sharp finite-T error terms, while also highlighting phenomena such as arbitrarily large holes in the visible point sets and connections to quasicrystal tilings like Amman–Beenker and Penrose.

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.

Figures (1)

• Figure 1: The Minkowski embedding of $\mathcal{O}_K$, where $K={\bf Q}(\sqrt{5})$ and $\lambda= \frac{1}{2}(1+\sqrt{5})$.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof