We study how the properties of a dilute Fermi gas with short-range interactions can be studied in terms of lattice models, in which the interaction can be restricted to electrons sitting on the same lattice site, as in the popular attractive Hubbard model. Our results, based on static and dynamical mean-field theory, show that the extrapolation to the zero-density limit, in which the effects of the lattice are bound to disappear, requires extreme care. In particular we identify a range of densities that can be used to reliably extrapolate to zero, and we show that this range becomes smaller if the correlation effects are more accurately taken into account.