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 \({\hat{\rho }}_{d}={\textrm{O}(d^2\log _2 d)}R\) 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 \({\hat{\rho }}_{d}={\textrm{O}(d\log _2 d)}R\) as \(d\rightarrow \infty \), independent of r.