  We study the properties of dipolar fermions trapped in one-dimensional
bichromatic optical lattices and show the existence of fractional topological
states in the presence of strong dipole-dipole interactions. We find some
interesting connections between fractional topological states in
one-dimensional superlattices and the fractional quantum Hall states: (i) the
one-dimensional fractional topological states for systems at filling factor
\nu=1/p have p-fold degeneracy, (ii) the quasihole excitations fulfill the same
counting rule as that of fractional quantum Hall states, and (iii) the total
Chern number of p-fold degenerate states is a nonzero integer. The existence of
crystalline order in our system is also consistent with the thin-torus limit of
the fractional quantum Hall state on a torus. The possible experimental
realization in cold atomic systems offers a new platform for the study of
fractional topological phases in one-dimensional superlattice systems.
