  The superfluid transition temperature $T_c$ of a unitary Fermi gas on a
three-dimensional isotropic lattice with an attractive on-site interaction is
investigated as a function of density $n$, from half filling down to $5.0\times
10^{-7}$ per unit cell, using a pairing fluctuation theory. We show that except
at very low densities ($n^{1/3} <0.2$), where $T_c/E_F$ is linear in $n^{1/3}$,
$T_c/E_F$ exhibits significant higher order nonlinear dependence on $n^{1/3}$.
Therefore, linear extrapolation using results at intermediate densities such as
in typical quantum Monte Carlo simulations leads to a significant underestimate
of the zero density limit of $T_c/E_F$. Our result, $T_c/E_F=0.256$, at $n=0$
is subject to reduction from particle-hole fluctuations and incoherent single
particle self energy corrections.
