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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
##### Keywords:
Koksma integral; mean value; 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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