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 $\hatρ_d$ is defined as the smallest positive number $ρ$ such that each Delone set with congruent clusters of radius $ρ$ 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 ${\widehat{\rho }}_d=\mathrm{O}(d^2\log d)R$ as $d\to \mathrm{\infty }$, 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 ${\widehat{\rho }}_d=\mathrm{O}(d\log d)R$ as $d\to \mathrm{\infty }$, independent of $r$.