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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:
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