Local Complexity of Delone Sets and

This paper characterizes when a Delone setX in Rn is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. For a Delone set X, let NX(T ) count the number of translation-inequivalent patches of radius T in X and let MX(T ) be the minimum radius such that every closed ball of radius MX(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 linear growth is equivalent to X being an ideal crystal. Explicitly, for NX(T ), if R is the covering radius of X then either NX(T ) is bounded or NX(T ) ≥ T/2R for all T > 0. The constant 1/2R in this bound is best possible in all dimensions. For MX(T ), either MX(T ) is bounded or MX(T ) ≥ 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 MX(T ) ≥ 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. AMS Subject Classification (2000): Primary: 52C23, 52C45 Secondary: 52C17