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

TL;DR

This paper establishes a new linear lower bound for the regularity radius of Delone sets in relation to their empty ball radius, advancing understanding of conditions for crystallinity in higher dimensions.

Contribution

It provides the first linear lower bound on the regularity radius in terms of dimension and empty ball radius for Delone sets, improving previous bounds.

Findings

Lower bound for regularity radius: rac{d}{d} imes 2R

Explicit constructions of non-regular Delone sets with equivalent clusters

Advancement in understanding local conditions for crystallinity

Abstract

The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set $X$ to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius $\hat{\rho}_d$ for Delone sets $X$ in terms of the radius $R$ of the largest "empty ball" for $X$. The present paper establishes the lower bound $\hat{\rho_d}\geq 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been $\hat{\rho}_d\geq 4R$ for $d\geq 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent $(2dR-\varepsilon)$-clusters, which are not regular systems.