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On the \(L^ 2\) discrepancy of distances of points from a finite sequence. (English) Zbl 0755.11022
Let \(\omega=(x_ n)_{n=1}^ \infty\) be a sequence in the unit interval and let \(\Omega=(| x_ m-x_ n|)_{m,n=1}^ \infty\) be the sequence consisting of all the distances \(| x_ m-x_ n|\) and which are ordered such that the first \(N^ 2\) terms are \((| x_ m-x_ n|)_{m,n=1}^ N\) for all \(N=1,2,\dots\). In Theorem 1 an explicit formula for the \(L^ 2\)-discrepancy of \(\Omega_ N\) in terms of integrals over “relative” discrepancies of \(\Omega_ N\) are established. Furthermore integral formulas for Stieltjes integrals with respect to “relative” discrepancies (= differences of empirical distributions and the uniform distribution) are shown. As an application the author obtains that \(\omega\) is uniformly distributed if and only if \(\Omega\) has the asymptotic distribution function \(2x-x^ 2\).
Reviewer: R.F.Tichy (Graz)

MSC:
11K06 General theory of distribution modulo \(1\)
65D30 Numerical integration
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References:
[1] VINOGRADOV I. M.: On fractional parts of integer polynomials. (Russian.) Izv. AN SSSR, 20, 1926, 585-600.
[2] VAN DER CORPUT J. G. PISOT C.: Sur la discrépance modulo un. Indag. Math., 1, 1939, pp. 143-152, 184-195, 260-269. · JFM 65.0170.02
[3] KUIPERS L., NIEDERREITER H.: Uniform Distribution of Sequences. J. Wiley, New York 1974. · Zbl 0281.10001
[4] STRAUCH O.: Some applications of Franel-Kluyver’s integral, II. Math. Slovaca, 39, 1989, No. 2, 127-140. · Zbl 0671.10002
[5] NIEDERREITER H.: Application of diophantine approximations to numerical integration. Diophantine Approximation and its Applications C. F. Osgood. Academic Press, New York 1973, pp. 129-199. · Zbl 0268.65014
[6] LeVEQUE W. J.: An inequality connected with Weyľs criterion for uniform distribution. Proc. Symp. Pure Math., Vol. VIII., American Math. Society, Providence 1965, p. 22-30.
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