Local Complexity of Delone Sets and Crystallinity

Lagarias, Jeffrey C. ; Pleasants, Peter A. B.

Canadian mathematical bulletin, Tome 45 (2002) no. 4, pp. 634-652

Résumé

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 ${{N}_{X}}\left( T \right)$ count the number of translation-inequivalent patches of radius $T$ in $X$ and let ${{M}_{X}}\left( T \right)$ be the minimum radius such that every closed ball of radius ${{M}_{X}}\left( T \right)$ 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 ${{N}_{X}}\left( T \right)$ , if $R$ is the covering radius of $X$ then either ${{N}_{X}}\left( T \right)$ is bounded or ${{N}_{X}}\left( T \right)\,\ge \,T/2R$ for all $T\,>\,0$ . The constant $1/2R$ in this bound is best possible in all dimensions.For ${{M}_{X}}\left( T \right)$ , either ${{M}_{X}}\left( T \right)$ is bounded or ${{M}_{X}}\left( T \right)\ge T/3$ 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 ${{M}_{X}}\left( T \right)\,\ge \,c\left( n \right)T$ for all $T\,>\,0$ , for a certain constant $c\left( n \right)$ which depends on the dimension $n$ of $X$ and is $>\,1/3$ when $n\,>\,1$ .

Détail

DOI : 10.4153/CMB-2002-058-0

Mots-clés : 52C23, 52C17, aperiodic set, Delone set, packing-covering constant, sphere packing