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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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