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Abstract
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Wall’s theorem on arithmetic progressions says that if
is normal,
then for any
,
is also normal. We examine a converse statement and show that if
is normal for periodic increasing
sequences
of asymptotic
density arbitrarily close to
,
then
is normal. We show this is close to sharp in the sense that there are numbers
that are not normal,
but for which
is normal along a large collection of sequences whose density is bounded a little away
from
.
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Keywords
normal number, decimal expansion
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Mathematical Subject Classification 2010
Primary: 11K16
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Milestones
Received: 3 January 2018
Revised: 25 September 2018
Accepted: 26 September 2018
Published: 30 July 2019
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