  The problem of finding the minimum-energy configuration of particles on a
lattice, subject to a generic short-ranged repulsive interaction, is studied
analytically. The study is relevant to charge ordered states of interacting
fermions, as described by the spinless Falicov-Kimball model. For a range of
particle density including the half-filled case, it is shown that the
minimum-energy states coincide with the large-U neutral ground state ionic
configurations of the Falicov-Kimball model, thus providing a characterization
of the latter as ``most homogeneous'' ionic arrangements. These obey
hierarchical rules, leading to a sequence of phases described by the Farey
tree. For lower densities, a new family of minimum-energy configurations is
found, having the novel property that they are aperiodic even when the particle
density is a rational number. In some cases there occurs local phase
separation, resulting in an inherent sensitivity of the ground state to the
detailed form of the interaction potential.
