On the discrepancy estimate of normal numbers

Abstract

1.1. A number α ∈ (0, 1) is said to be normal to base q if in the q-ary expansion of α, α = .d1d2 . . . (di ∈ ∆ = {0, 1, . . . , q − 1}, i = 1, 2, . . .), each fixed finite block of digits of length k appears with an asymptotic frequency of q−k along the sequence (di)i≥1. Normal numbers were introduced by Borel (1909). 1.1.1. Let (xn)n≥1 be an arbitrary sequence of real numbers. The quantity (1) D(N) = D(N, (xn)n≥1) = sup γ∈(0,1] |#{0 ≤ n < N | {xn} < γ}/N − γ|