  I study the leading effects of discretization errors on the low energy part
of the spectrum of the Hermitian Wilson-Dirac operator in infinite volume. The
method generalizes that used to study the spectrum of the Dirac operator in the
continuum, and uses partially quenched chiral perturbation theory for Wilson
fermions. The leading-order corrections are proportional to a^2 (a being the
lattice spacing). At this order I find that the method works only for one
choice of sign of one of the three low energy constants describing
discretization errors. If these constants have the relative magnitudes expected
from large N_c arguments, then the method works if the theory has an Aoki phase
for m of order a^2, but fails if there is a first-order transition. In the
former case, the dependence of the gap and the spectral density on m and a^2
are determined. In particular, the gap is found to vanish more quickly as
m_pi^2-> 0 than in the continuum. This reduces the region where simulations are
safe from fluctuations in the gap.
