Most numbers are not normal

Abstract

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers x ∈ (0, 1] with the following property is comeager: for all integers b ≥ 2 and k ≥ 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.