# 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. Local complexity of Delone sets and crystallinity Canadian Mathematical Bulletin-bulletin Canadien De Mathematiques   Check publisher's open access policy 0008-4395 2002-12 Article (original research) 10.4153/CMB-2002-058-0 45 4 634 652 19 N. YuiJ. Lewis Montreal, Canada Societe mathematique du Canada 2002 engfre C1230111 Geometry780101 Mathematical sciences 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. MathematicsGeometric-modelsQuasi-crystalsTilings C1

 Document type: Journal Article Article (original research) School of Mathematics and Physics

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