Without doing much work, one can guess that the electron-electron interactions will scale like the inverse of the average electron-electron separation and hence as formula_9 (since the Coulomb interaction goes like one over distance between charges) so that if we view the interactions as a small correction to the kinetic energy, we are describing the limit of small formula_8 (i.e. formula_11 being larger than formula_12) and hence high electron density. Unfortunately, real metals typically have formula_13 between 2-5 which means this picture needs serious revision.