Local complexity of Delone sets and crystallinity

Lagarias, J. C. and Pleasants, P. A. B. (2002) Local complexity of Delone sets and crystallinity. Canadian Mathematical Bulletin-bulletin Canadien De Mathematiques, 45 4: 634-652. doi:10.4153/CMB-2002-058-0


Author Lagarias, J. C.
Pleasants, P. A. B.
Title Local complexity of Delone sets and crystallinity
Journal name Canadian Mathematical Bulletin-bulletin Canadien De Mathematiques   Check publisher's open access policy
ISSN 0008-4395
Publication date 2002-12
Sub-type Article (original research)
DOI 10.4153/CMB-2002-058-0
Volume 45
Issue 4
Start page 634
End page 652
Total pages 19
Editor N. Yui
J. Lewis
Place of publication Montreal, Canada
Publisher Societe mathematique du Canada
Collection year 2002
Language eng
fre
Subject C1
230111 Geometry
780101 Mathematical sciences
Abstract This paper characterizes when a Delone set X in R-n is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set X, let N-X (T) count the number of translation-inequivalent patches of radius T in X and let M-X (T) be the minimum radius such that every closed ball of radius M-X(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 sufficiently slow linear growth is equivalent to X being an ideal crystal. Explicitly, for N-X (T), if R is the covering radius of X then either N-X (T) is bounded or N-X (T) greater than or equal to T/2R for all T > 0. The constant 1/2R in this bound is best possible in all dimensions. For M-X(T), either M-X(T) is bounded or M-X(T) greater than or equal to 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-X(T) greater than or equal to c(n)T for all T > 0, for a certain constant c(n) which depends on the dimension n of X and is > 1/3 when n > 1.
Keyword Mathematics
Geometric-models
Quasi-crystals
Tilings
Q-Index Code C1

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
 
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Created: Tue, 14 Aug 2007, 17:37:51 EST