Abelian-normal decimal expansions

Abstract

A landmark paper on normal numbers due to Champernowne [J. London Math. Soc., 1933] provided the first explicit construction of a normal number. Since many research works have concerned normality-preserving selection rules and operations on the sequence of digits of a given normal number, this leads us to introduce the concept of an abelian-normal number, in such a way so that subwords that are equivalent up to permutations are counted as being equivalent, and so that the number of occurrences of equivalent subwords appearing, up to a given point in a decimal expansion, is then normalized by the size of the associated equivalence class. This may be thought of as giving rise to an interdisciplinary area linking combinatorics on words and probabilistic number theory. Since every normal number is abelian-normal, this raises the question as to whether or not there exists an abelian-normal number that is not normal. We succeed in constructing such a constant in an explicit way.