  We investigate the interplay between disorder and interactions in a Bose gas
on a lattice in presence of randomly localized impurities. We compare the
performance of two theoretical methods, the simple version of multi-orbital
Hartree-Fock and the common Gross-Pitaevskii approach, showing how the former
gives a very good approximation to the ground state in the limit of weak
interactions, where the superfluid fraction is small. We further prove
rigorously that for this class of disorder the fractal dimension of the ground
state d* tends to the physical dimension in the thermodynamic limit. This
allows us to introduce a quantity, the fractional occupation, which gives
insightful information on the crossover from a Lifshits to a Bose glass.
Finally, we compare temperature and interaction effects, highlighting
similarities and intrinsic differences.
