Finding the optimal arrangements (ground states) of an infinite number of interacting particles is a notoriously difficult problem. Here we will show how Fourier analysis is helpful, if the interaction is an integrable pair potential. If the Fourier transform of the interaction is nonnegative on a compact support, a complete description of ground states at high densities is possible. At a threshold density a unique ground state appears as the reciprocal lattice of the close-packed (Bravais) lattice (hence, the basic-centred cubic lattice in 3 dimensions). Above this density there is a continuum of ground state configurations. If the interaction’s Fourier transform is strictly positive, as for the Gaussian, then in any dimension the particle distribution tends to be uniform as the density goes to infinity. If on the other hand the Fourier transform is partially negative, the asymptotic distribution is non-uniform. Examples will be given of interactions in any dimension that have no Bravais-lattice ground state at high enough density.