Wall’s theorem on arithmetic progressions says that if
is normal,
then for any
,
is also normal. We examine a converse statement and show that if
is normal for periodic increasing
sequences
of asymptotic
density arbitrarily close to
,
then
is normal. We show this is close to sharp in the sense that there are numbers
that are not normal,
but for which
is normal along a large collection of sequences whose density is bounded a little away
from
.