Local Conditions of Crystal Structures

We will give an overview of the Local Theory of regular systems and also its connection with studies of quasicrystals and of arbitrary Delone/Delaunay set. The Local Theory of regular systems relates to the foundations of Geometric Crystallography.
The mathematical model of an ideal crystal (its atomic structure) is a discrete subset $X$ in a finite-dimensional Euclidean space that is invariant with respect to some crystallographic group $G$ of isometries of the Euclidean space. In other words, a crystal $X$ is the union of a finite number of $G$-orbits.
A single-point orbit is termed a regular system. Our attention will be focused on the lower and upper bounds for the regularity radius, which is the minimum size of clusters whose pairwise equivalence at all points of a Delone set provides the regularity of the set.