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Principles of a new method in the study of irregularities of distribution. (English) Zbl 0721.11028
A new method in the theory of irregularities of distribution is presented; the method is mainly based on ideas of integral geometry. Define a signed measure $$\nu =\nu^{+}-\nu^{-}$$ where $$\nu^{+}$$ is the normalized Lebesgue measure (on some region in Euclidean space) and $$\nu^{-}$$ is the atomic measure that assigns weight $$1/n$$ to given points $$p_k$$ for $$k\leq n$$. The problem discussed in the paper is to find lower bounds for the discrepancy
$D(\nu)=\sup | \nu(H)|,$ where $$H$$ ranges over all open half-spaces. The main idea is to consider functionals of the type
$I^\alpha(\nu)=\iint | p - q|^\alpha\, d\nu(p)\,d\nu(q).$ For various choices of $$\alpha$$ and $$\nu$$ the functional $$I^\alpha$$ has had important applications, certainly for $$\alpha <0$$ in potential theory.
The present work is only concerned with $$0<\alpha <2$$ and the following fundamental estimate is proved:
Theorem A. Suppose that $$M$$ is a convex $$t$$-dimensional hypersurface embedded in a Euclidean space $$E^{t+1}$$. Let $$K$$ be any set of unit $$t$$-measure lying on $$M$$ and let $$p_1,\dots,p_n$$ be arbitrary points of $$M$$. Then for $$0<\alpha <2$$
$n^2\, | I^\alpha(\nu)| > c(t,\alpha)n^{1-\alpha /t}, \tag{1}$ where the constant $$c$$ is independent of $$\nu$$, $$M$$ and $$K$$.
For the special case of the $$t$$-sphere J. Beck [Mathematika 31, 33–41 (1984; Zbl 0553.51013)] has established a result that implies (1) by means of Fourier transformation techniques; G. Wagner has recently established a result equivalent to (1) for $$t=2$$ by means of spherical harmonics.
Using ideas from integral geometry the following second main result follows from Theorem A:
Theorem B. Let $$\nu$$ be as in Theorem A and assume that $$\nu$$ is supported by a disk of radius $$r$$. Then
$n\,D(\nu)>c_tR^{-1/2}n^{1/2-1/2d}, \tag{2}$ where the constant $$c_t$$ depends only on $$t$$.
In the case of the $$t$$-sphere (2) was obtained by J. Beck applying his Fourier transform method, a little weaker result is due to W. M. Schmidt by means of his integral equation method; cf. J. Beck and W. W. L. Chen [Irregularities of distribution. Cambridge etc.: Cambridge University Press (1987; Zbl 0617.10039)].

##### MSC:
 11K38 Irregularities of distribution, discrepancy
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##### References:
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