On the regularity radius of Delone sets in $\mathbb{R}^3$

TL;DR

This paper proves that in three-dimensional Euclidean space, if all clusters within a radius of 10 times the Delone set parameter are equivalent, then the set is a regular system, completing the proof of the upper bound for the regularity radius.

Contribution

It completes the proof of the upper bound for the regularity radius of Delone sets in 3D by addressing the remaining cases.

Findings

Established the upper bound $ ho_3 \,\leq\, 10R$ for the regularity radius.

Demonstrated that equivalence of all $10R$-clusters implies the set is a regular system.

Filled in the missing cases in the proof of the regularity radius bound.

Abstract

We complete the proof of the upper bound $\hat\rho_3\leq 10R$ for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all $10R$-clusters of a Delone set $X$ with parameters $(r,R)$ are equivalent, then $X$ is a regular system.