  In the dilute limit, the properties of fermionic lattice models with
short-range attractive interactions converge to those of a dilute Fermi gas in
continuum space. We investigate this connection using mean-field and we show
that the existence of a finite lattice spacing has consequences down to very
small densities. In particular we show that the reduced translational
invariance associated to the lattice periodicity has a pivotal role in the
finite-density corrections to the universal zero-density limit.
  For a parabolic dispersion with a sharp cut-off, we provide an analytical
expression for the leading-order corrections in the whole BCS-BEC crossover.
These corrections, which stem only from the unavoidable cut-off, contribute to
the leading-order corrections to the relevant observables. In a generic lattice
we find a universal power-law behavior $n^{1/3}$ which leads to significant
corrections already for small densities. Our results pose strong constraints on
lattice extrapolations of dilute Fermi gas properties.
