  Recently it has been suggested that fermions whose hopping amplitude is
quenched to extremely low values provide a convenient source of local disorder
for lattice bosonic systems realized in current experiment on ultracold atoms.
Here we investigate the phase diagram of such systems, which provide the
experimental realization of a Bose-Hubbard model whose local potentials are
randomly extracted from a binary distribution. Adopting a site-dependent
Gutzwiller description of the state of the system, we address one- and
two-dimensional lattices and obtain results agreeing with previous findings, as
far as the compressibility of the system is concerned. We discuss the expected
peaks in the experimental excitation spectrum of the system, related to the
incompressible phases, and the superfluid character of the {\it partially
compressible phases} characterizing the phase diagram of systems with binary
disorder. In our investigation we make use of several analytical results whose
derivation is described in the appendices, and whose validity is not limited to
the system under concern.
