  Let $G$ be a connected reductive group over a totally real field $F$ which is
compact modulo center at archimedean places. We find congruences modulo an
arbitrary power of p between the space of arbitrary automorphic forms on
$G(\mathbb A_F)$ and that of automorphic forms with supercuspidal components at
p, provided that p is larger than the Coxeter number of the absolute Weyl group
of $G$. We illustrate how such congruences can be applied in the construction
of Galois representations.
  Our proof is based on type theory for representations of p-adic groups,
generalizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to
general reductive groups. We exhibit a plethora of new supercuspidal types
consisting of arbitrarily small compact open subgroups and characters thereof.
We expect these results of independent interest to have further applications.
For example, we extend the result by Emerton--Pa\v{s}k\=unas on density of
supercuspidal points from definite unitary groups to general $G$ as above.
