A metrical result on the discrepancy of (nα)

In the following let Ω be the set of irrational numbers in the interval [0,1] and let λ be Lebesgue measure restricted to Ω. For any real number x, let {x} = x - [x] be the fractional part of x. Let N be anatural number and let α e Ω. Thenis known as the discrepancy of the sequence (nα)n>1 modulo 1; here c[x, y) denotes the characteristic function of the interval [x, y).