Sequences with bounded logarithmic discrepancy

Abstract

Let {α} denote the fractional part of the real number α. Write χ(x, y) = 1 for {x} < y and χ(x, y) = 0 otherwise. A real sequence (xn) is uniformly distributed (mod 1) ifIt is a consequence of (1·1) that DN = o(N),Whereis the discrepancy of the sequence (xn).