  Wall's theorem on arithmetic progressions says that if $0.a_1a_2a_3\dots$ is
normal, then for any $k,\ell\in \mathbb{N}$, $0.a_ka_{k+\ell}a_{k+2\ell}\dots$
is also normal. We examine a converse statement and show that if
$0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal for periodic increasing sequences
$n_1<n_2<n_3<\dots$ of asymptotic density arbitrarily close to $1$, then
$0.a_1a_2a_3\dots$ is normal. We show this is close to sharp in the sense that
there are numbers $0.a_1a_2a_3\dots$ that are not normal, but for which
$0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal along a large collection of sequences
whose density is bounded a little away from $1$.
