  Pair interactions whose Fourier transform is nonnegative and vanishes above a
wave number K_0 are shown to give rise to periodic and aperiodic infinite
volume ground state configurations (GSCs) in any dimension d. A typical three
dimensional example is an interaction of asymptotic form cos(K_0 r)/r^4. The
result is obtained for densities rho >= rho_d where rho_1=K_0/2pi,
rho_2=(sqrt{3}/8)(K_0/pi)^2 and rho_3=(1/8sqrt{2})(K_0/pi)^3. At rho_d there is
a unique periodic GSC which is the uniform chain, the triangular lattice and
the bcc lattice for d=1,2,3, respectively. For rho>rho_d the GSC is nonunique
and the degeneracy is continuous: Any periodic configuration of density rho
with all reciprocal lattice vectors not smaller than K_0, and any union of such
configurations, is a GSC. The fcc lattice is a GSC only for rho>=(1/6
sqrt{3})(K_0/pi)^3.
