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

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 $\widehat{{\rho }_d}\ge 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been ${\hat{\rho }}_d\ge 4R$ for $d\ge 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent $(2dR-\epsilon )$-clusters, which are not regular systems.