Equidistribution Mod $1$ And Normal Numbers

Abstract

Let α = 0.a 1 a 2 a 3 . . . be an irrational number in base b > 1, where 0 ≤ a i < b. The number α ∈ (0, 1) is a normal number if every block (a n+1 a n+2 . . . a n+k ) of k digits occurs with probability 1/b k . A conditional proof of the normality of the real number π in base 10 is presented in this note.