Local Complexity of Delone Sets and Crystallinity

http://dx.doi.org/10.4153/CMB-2002-058-0
Canad. Math. Bull. 45(2002), 634-652

  Published:2002-12-01
 Printed: Dec 2002

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 $n>1$.