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

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.