#### Vol. 300, No. 2, 2019

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

### Joseph Vandehey

Vol. 300 (2019), No. 2, 499–509
##### Abstract

Wall’s theorem on arithmetic progressions says that if $0.{a}_{1}{a}_{2}{a}_{3}\dots$ is normal, then for any $k,\ell \in ℕ$, $0.{a}_{k}{a}_{k+\ell }{a}_{k+2\ell }\dots$ 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 $1$.

##### Keywords
normal number, decimal expansion
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