  We investigate the density of states (DOS) near the Fermi energy of
one-dimensional spin-polarized electron systems in the quantum regime where the
localization length is comparable to or larger than the inter-particle
distance. The Wigner lattice gap of such a system, in the presence of weak
disorder, can occur precisely at the Fermi energy, coinciding with the Coulomb
gap in position. The interplay between the two is investigated by treating the
long-range Coulomb interaction and the random disorder potential in a
self-consistent Hartree-Fock approximation. The DOS near the Fermi energy is
found to be well described by a power law whose exponent decreases with
increasing disorder strength.
