## School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Article

6-2023

#### Abstract

Delone sets are discrete point sets X in Rd 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 ρ^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 ρ^d=O(d2logd)R as d→∞, 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 ρ^d=O(dlogd)R as d→∞, independent of r.

Delone sets are discrete point sets X in Rd 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 ρ^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 ρ^d=O(d2logd)R as d→∞, 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 ρ^d=O(dlogd)R as d→∞, independent of r.