Bounds for the Regularity Radius of Delone Sets

Delone sets are discrete point sets $X$ in $\mathbb{R}^d$ characterized by parameters $(r,R)$, where (usually) $2r$ is the smallest inter-point distance of $X$, and $R$ is the radius of a largest ``empty ball" that can be inserted into the interstices of $X$. The regularity radius $\hat{\rho}_d$ is defined as the smallest positive number $\rho$ such that each Delone set with congruent clusters of radius $\rho$ is a regular system, that is, a point orbit under a crystallographic group. We discuss two conjectures on the growth behavior of the regularity radius. Our ``Weak Conjecture" states that $\hat{\rho}_{d}={{\rm O}(d^2\log d)}R$ as $d\rightarrow\infty$, independent of~$r$. This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius $2r$ and those with full-dimensional sets of $d$-reachable points. We also offer support for the plausibility of a ``Strong Conjecture", stating that $\hat{\rho}_{d}={{\rm O}(d\log d)}R$ as $d\rightarrow\infty$, independent of $r$.