School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Document Type
Article
Publication Date
6-22-2024
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(d2log2d)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(dlog2d)R as d→∞ , independent of r.
Recommended Citation
Dolbilin, Nikolay, Alexey Garber, Egon Schulte, and Marjorie Senechal. "Bounds for the regularity radius of Delone sets." Discrete & Computational Geometry (2024): 1-17. https://doi.org/10.1007/s00454-024-00666-6
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication Title
Discrete & Computational Geometry
DOI
https://doi.org/10.1007/s00454-024-00666-6
Comments
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.