Quasicrystal

Definition

Let $n \ge 1$ and let $\Lambda \subset \R^n$ be a set of points. $\Lambda$ is a quasicrystal (in the mathematical sense) if the following hold.

1. $\Lambda$ is uniformly discrete: there is $r > 0$ with $\abs{x - y} \ge r$ for all distinct $x, y \in \Lambda$.
2. $\Lambda$ is relatively dense: there is $R > 0$ such that every closed ball of radius $R$ meets $\Lambda$.
3. $\Lambda$ has pure point diffraction: the autocorrelation measure $\gamma = \lim_{s \to \infty} \frac{1}{\operatorname{vol}(B_s)} \sum_{x, y \in \Lambda \cap B_s} \delta_{x - y}$ exists, and its Fourier transform $\hat\gamma$, the diffraction measure, is a pure point measure.
4. $\Lambda$ is not crystallographic: there is no Bravais lattice $L \subset \R^n$ with $\Lambda + L = \Lambda$.

Here $B_s$ is the ball of radius $s$ centred at $0$, and the limit is in the vague topology of measures. A set with clauses 1 and 2 is a Delone set.