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 Πi(x, n) denote the frequency of the digit i among the first n N -adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the frequencies diverge as n → ∞. In this paper we provide a substantial strengthening of this result. Namely, we show that for a typical x ∈ [0, 1] any regular linear average of the sequence (Πi(x, n))n also diverges spectacularly.