Bounds for the Regularity Radius of Delone Sets

Delone sets are discrete point sets X in Rᵈ Rd characterized by parameters ( r , R ), where (usually) 2 r 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 ρ ̂d ρ^d is defined as the smallest positive number ρ ρ such that each Delone set with congruent clusters of radius ρ 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 ρ ̂d=O(d²log ₂ d)R ρ^d=O(d2log2d)R as d→ ∞ d→∞ , independent of r . This is verified in the paper for two important subfamilies of Delone sets: those with full-dimensional clusters of radius 2 r and those with full-dimensional sets of d -reachable points. We also offer support for the plausibility of a “Strong Conjecture”, stating that ρ ̂d=O(dlog ₂ d)R ρ^d=O(dlog2d)R as d→ ∞ d→∞ , independent of r .