Wall’s theorem on arithmetic progressions says that if
$0.{a}_{1}{a}_{2}{a}_{3}\dots $ is normal,
then for any
$k,\ell \in \mathbb{N}$,
$0.{a}_{k}{a}_{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
${n}_{1}<{n}_{2}<{n}_{3}<\cdots \phantom{\rule{0.3em}{0ex}}$ of asymptotic
density arbitrarily close to
$1$,
then
$0.{a}_{1}{a}_{2}{a}_{3}\dots $
is normal. We show this is close to sharp in the sense that there are numbers
$0.{a}_{1}{a}_{2}{a}_{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
$1$.
