An improvement of an inequality of Koksma
Let \(\alpha\in\mathbb{R}\), \(0\leq x<y\leq 1\); \(q_ 1,q_ 2,\dots,q_ N\) positive integers; \(w_ N(\alpha)\) the finite sequence \(\{q_ n\alpha\}\); \(n=1,\dots,N\) and \(R_ N(x,y,\alpha)\) the discrepancy function of \(w_ N(\alpha)\) for the interval \([x,y)\). It was shown by Koksma that \[ I:=\int_ 0^ 1 R_ N^ 2(x,y,\alpha) d\alpha\leq{1\over3}\sum_{m,n=1}^ N {{(q_ m,q_ n)}^ 2 \over q_ m q_ n}. \] The author gives an explicit representation of the left hand integral to show that \[ I\leq{1\over 4}\sum_{m,n=1}^ N {(q_ m,q_ n)^ 2 \over q_ m q_ n}. \] The constant \(1/4\) is best possible. Further it is shown that \[ 2\iint_{0\leq x<y\leq 1} \int_ 0^ 1 R_ N^ 2(x,y,\alpha) d\alpha dx dy={1\over 6} \sum_{{m,n=1}\atop{q_ m=q_ n}}^ N 1. \]
scientific article; zbMATH DE number 57236