Towards a sharp converse of Wall's theorem on arithmetic progressions

Wall’s theorem on arithmetic progressions says that if $0.a_1{a}_2{a}_3\dots$ is normal, then for any $k,\mathrm{\ell }\in \mathbb{N}$, $0.a_k{a}_{k+\mathrm{\ell }}{a}_{k+2\mathrm{\ell }}\dots$ 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 $0.a_1{a}_2{a}_3\dots$ 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 $1$.