  In a previous letter, L. Baguet et al., (Phys. Rev. Lett. {\bf 111}, 166402
(2013)), we presented the ground state phase diagram of the homogeneous
electron gas in three dimensions within the Hartree-Fock approximation yielding
incommensurate crystal states at high density. Here, we analyze the properties
of these solutions. In particular, at high density we find universal behavior
of the incommensurate crystal strongly supporting the existence of a spin
density wave ground state.
