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1. CMB 2002 (vol 45 pp. 634)

Lagarias, Jeffrey C.; Pleasants, Peter A. B.
 Local Complexity of Delone Sets and Crystallinity This paper characterizes when a Delone set \$X\$ in \$\mathbb{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) \ge 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)\ge 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)\ge 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\$. Keywords:aperiodic set, Delone set, packing-covering constant, sphere packingCategories:52C23, 52C17