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.a1a2a3 is normal, then for any k, , 0.akak+ak+2 is also normal. We examine a converse statement and show that if 0.an1an2an3 is normal for periodic increasing sequences n1 < n2 < n3 < of asymptotic density arbitrarily close to 1, then 0.a1a2a3 is normal. We show this is close to sharp in the sense that there are numbers 0.a1a2a3 that are not normal, but for which 0.an1an2an3 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
Authors
Joseph Vandehey
Department of Mathematics
The Ohio State University
Columbus, OH
United States