A Typical Number is Extremely Non-Normal

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.