What are some properties of Delone sets that come from Barlow packings of spheres? Given a Barlow packing of $\mathbb{R}^n$ by balls with at most a finite number of different radii, the centers of the balls will form a Delone set in $\mathbb{R}^n.$ 
For a highest density sphere packing, or at least a Barlow packing of highest density among Barlow packings, must the corresponding Delone set be a Meyer set? A Patterson set? In the cases of dimensions 2 and 3 where the optimal packing using a single radius is known, the sets can be chosen to be lattices. 
I looked for papers exploring this connection and could only find this one https://www-fourier.ujf-grenoble.fr/PUBLIS/publications/REF_678.pdf, which is good, but doesn't address the big picture questions above. 
EDIT: As the comments indicate, we should restrict to Barlow packings. In this case the Delone sets always appear to be of finite local complexity. 
 A: If you consider the Barlow lattices, these are obtained by taking
planar hexagonal packings of spheres, and nesting them together
in layers. For each layer, there are two possible ways of placing
the layer above it:

(source: cnx.org)
Let's normalize the Barlow lattices to have one common layer (the A-layer).
The heights of the (centers of spheres of the) other layers will be multiples of the height of a
tetrahedron, so are the same for each Barlow lattice. The centers of the vertices of the next
layer above have two possibilities, one with projection given by the B-circles, and one
given by the C-circles (these are symmetric under a transformation
preserving A-circles, so let's assume the next layer is the B-circles).
Then the next layer can have projection either the C-circles or the A-circles
(the diagram corresponds to the face-centered cubic lattice, so the pattern
there is ABCABC...). Thus, we see that for any Barlow lattice D, the
differences $D-D$ will be a subset of the superposition of 3 face-centered cubic
lattices which have the A-,B-,C-layers at each level, and thus will be a discrete set. So it will be a Meyer set (I'm
going by the definition of a Meyer set as a discrete set $M$ such that $M-M$ is
also Delone).
