The discrepancy of some real sequences - Haili, H. Kamarul; Nair, R. (MATHEMATICA SCANDINAVICA, 2003)
Let $(λ_n)_{n≥ 0}$ be a sequence of real numbers such that there exists $δ > 0$ such that $|λ_{n+1} - λ_n| ≥ δ , n = 0,1,...$. For a real number $y$ let $\{ y \}$ denote its fractional part. Also, for the real number $x$ let $D(N,x)$ denote the discrepancy of the numbers $\{ λ _0 x \}, … , \{ λ _{N-1} x \}$. We show that given $ε > 0$, 9774 D(N,x) = o ( N^{-\frac{1}{2}}(\log N)^{\frac{3}{2} + ε})9774 almost everywhere with respect to Lebesgue measure.