  We calculate the phase diagram of the Bose-Fermi Hubbard model on the 3d
cubic lattice at fermionic half filling and bosonic unit filling by means of
single-site dynamical mean-field theory. For fast bosons, this is equivalent to
the Cooper problem in which the bosons can induce s-wave pairing between the
fermions. We also find miscible superfluid and canted supersolid phases
depending on the interspecies coupling strength. In contrast, slow bosons favor
fermionic charge density wave structures for attractive fermionic interactions.
These competing instabilities lead to a rich phase diagram within reach of cold
gas experiments.
