A Typical Number is Extremely Non-Normal

Abstract

Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1,..., N − 1}, let Π i (x, n) denote the frequency of the digit i among the first nN-adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the sequence of digit frequencies diverges as n →∞. In this paper we show that for any regular linear transformation T there exists a residual set of points x ∈ [0,1] such that the T -averaged version of the sequence (Π i (x, n)) n also diverges significantly.

Most numbers are not normal

“…We extend the main results contained in [15,27]. To this aim, let A = (a n,i : n, i ∈ N) be a regular matrix, that is, an infinite real-valued matrix such that, if z = (z n ) is a R d -valued sequence convergent to η, then A n z := i a n,i z i exists for all n ∈ N and lim n A n z = η, see e.g.…”

“…The main result in [27] corresponds to the case I = Fin and k = 1, although with a different proof; cf. also Example 4•10 below.…”

“…In addition, the main result in [27] states that Theorem 4•2, specialised to the case I = Fin and k = 1, can be further strengtened so that the set {x ∈ (0, 1] : n,i with entries in {0, 1} so that a (y)…”

“…In addition, the main result in [27] states that Theorem 4•2, specialised to the case I = Fin and k = 1, can be further strengtened so that the set {x ∈ (0, 1] : 1 b (x, Fin, A) ⊇ 1 b for all b ≥ 2 and all regular A} is comeager. Taking into account the argument in the proof of Theorem 4•7, this would imply that the set {x ∈ (0, 1] : 1 b (x, Fin, A) = 1 b for all b ≥ 2 and all nonnegative regular A} (7) should be comeager.…”