Local Complexity of Delone Sets and Crystallinity
| dc.creator | Lagarias, J. C. | |
| dc.creator | Pleasants, P. A. B. | |
| dc.date | 2001-05-11 | |
| dc.date.accessioned | 2026-07-07T04:41:39Z | |
| dc.date.available | 2026-07-07T04:41:39Z | |
| dc.description | This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to X being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in this bound is best possible in all dimensions. For M(T), either M(T) is bounded or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1. | |
| dc.description | 26 pages. Uses latexsym and amsfonts packages | |
| dc.identifier | https://arxiv.org/abs/math/0105088 | |
| dc.identifier | http://arxiv.org/abs/math/0105088 | |
| dc.identifier | Canad. Math. Bull. Vol 48 (4), 2002 pp. 634-652 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61447 | |
| dc.subject | Metric Geometry | |
| dc.subject | 52C23 52C45 (Primary), 52C17 (Secondary) | |
| dc.title | Local Complexity of Delone Sets and Crystallinity | |
| dc.type | text |