The coincidence problem for shifted lattices and crystallographic point packings

A coincidence site lattice is a sublattice formed by the intersection of a lattice $\mathrm{\Gamma }$ in ${\mathbb{R}}^d$ with the image of $\mathrm{\Gamma }$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of $\mathrm{\Gamma }$. Here, we consider the more general case allowing any affine isometry. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, we discuss the shifted square lattice and the diamond packing in detail.