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An improvement of an inequality of Koksma. (English) Zbl 0755.11023
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 \[ 2\iint_{0\leq x<y\leq 1} \int_ 0^ 1 R_ N^ 2(x,y,\alpha) d\alpha dx dy={1\over 6} \sum_{{m,n=1}\atop{q_ m=q_ n}}^ N 1. \]

MSC:
11K31 Special sequences
11K38 Irregularities of distribution, discrepancy
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References:
[1] Kuipers, L.; Niederreiter, H., Uniform distribution of sequences, (1974), John Wiley & Sons New York · Zbl 0281.10001
[2] Koksma, J.F., On a certain integral in the theory of uniform distribution, Indag. math, 13, 285-287, (1951) · Zbl 0043.27701
[3] Porubský, Sˇ.; Sˇalát, T.; Strauch, O., Transformations that preserve uniform distribution, Acta arith., XLIX, 459-479, (1988) · Zbl 0656.10047
[4] Leveque, W.J., An inequality connected with Weyl’s criterion for uniform distribution, (), 22-30
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