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

Abstract

We complete the proof of the upper bound ${\hat{\rho }}_3\le 10R$ 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.

References in corpus (1)

• On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets