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

Joseph Vandehey

Vol. 300 (2019), No. 2, 499–509

DOI: 10.2140/pjm.2019.300.499

Abstract

Wall’s theorem on arithmetic progressions says that if $0.a_1{a}_2{a}_3\dots$ is normal, then for any $k,\ell \in \mathbb{N}$, $0.a_k{a}_{k+\ell }{a}_{k+2\ell }\dots$ 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 $n_1<n_2<n_3<\cdots$ 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

Mathematical Subject Classification 2010

Primary: 11K16

Milestones

Received: 3 January 2018

Revised: 25 September 2018

Accepted: 26 September 2018

Published: 30 July 2019