Most numbers are not normal

TL;DR

This paper demonstrates that, from a topological perspective, most numbers in (0,1] are not normal, meaning their digit frequency sequences have complex accumulation behaviors in all bases.

Contribution

It extends Olsen's result by showing a stronger form of non-normality for most numbers and offers a streamlined proof along with analogues in analytic P-ideals and regular matrices.

Findings

Most numbers are not normal in a strong topological sense.

The set of such numbers is comeager in (0,1].

The frequency sequences have complex accumulation point structures.

Abstract

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$, the sequence of vectors made by the frequencies of all possibile strings of length $k$ in the $b$-adic representation of $x$ has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen in [Math. Proc. Cambridge Philos. Soc. 137 (2004), 43--53]. We provide analogues in the context of analytic P-ideals and regular matrices.