2023/06/19 by Nikolay Dolbilin, Alexey Garber, Dolbilin, Nikolay +5
Biochemistry, Genetics and Molecular Biology · Computer Science · Materials Science · #Advanced Mathematical Modeling in Engineering #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #RNA Research and Splicing
paper · pdf · doi:10.48550/arxiv.2306.11127
openalex publication_date 2023/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ρd 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 ρd=\rm O(d2log d)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 ρd=\rm O(dlog d)R as d→∞, independent of r.