  Spectral functions are important quantities that contain a wealth of
information about the quasiparticles of a system, and that can also be measured
experimentally. For systems with electron-phonon coupling, good approximations
for the spectral function are available only in the Migdal limit (at Fermi
energies much larger than the typical phonon frequency, $E_F\gg \Omega$,
requiring a large carrier concentration $x$) and in the single polaron limit
(at $x=0$). Here we show that the region with $x\ll 1$ ($E_F <\Omega$) can also
be reliably investigated with the Momentum Average (MA) variational
approximation, which essentially describes the formation of a polaron above an
inert Fermi sea. Specifically, we show that for the one-dimensional spinless
Holstein model, the MA spectral functions compare favorably with those
calculated using variationally exact density matrix renormalization group
simulations (DMRG) evaluated directly in frequency-space, so long as $x<0.1$
and the adiabaticity ratio $\Omega/t>0.5$. Unlike in the Migdal limit, here
'polaronic physics' emerges already at moderate couplings. The relevance of
these results for a spinful low-$x$ metal is also discussed.
