Superimposed particles in 1D ground states

For a class of nonnegative, range-1 pair potentials in one dimensional continuous space we prove that any classical ground state of lower density >=1 is a tower-lattice, i.e., a lattice formed by towers of particles the heights of which can differ only by one, and the lattice constant is 1. The potential may be flat or may have a cusp at the origin, it can be continuous, but its derivative has a jump at 1. The result is valid on finite intervals or rings of integer length and on the whole line.