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Irregularities of distribution. X. (English) Zbl 0373.10020
Number Theory and Algebra. Collect. Pap. dedic. H. B. Mann, A. E. Ross, O. Taussky-Todd, 311-329 (1977).
Part IX, cf. Acta Arith. 27, 385–396 (1975; Zbl 0274.10038).
The title refers to irregularities of the distribution of an arbitrary set of $$N$$ points in the $$k$$-dimensional unit cube $$U^k$$ where $$k>1$$. If $$U^k$$ consists of points $$\underline y=(y_1,\dots, y_k)$$ with $$0\leq y_i<1$$, and if $$\underline x$$ lies in $$U^k$$, put $$Z(\underline x)$$ for the number of the given $$N$$ points which lie in the box $$0\leq y_i<1x_i$$ $$(i=1,\dots, k)$$. The irregularities of the distribution may be measured in various ways by the behaviour of the function $$D(\underline x)=Z(\underline x)-Nx_1\ldots x_k$$. If $$\| D\|_p$$ denotes the $$L^p$$-norm $$(\int| D|^p\,dx)^{1/p}$$, then it is shown that for $$p>1$$ we have
$\| D\|_p>c_1(k,p)(\log N)^{(k-1)/2}.$
The proof is based on a method of K. F. Roth [Mathematika, Lond. 1, 73–79 (1954; Zbl 0057.28604)] who did the case $$p=2$$, but in general additional difficulties occur in the estimation of certain moments. It is also shown that for $$p=1$$ we have
$\| D\|_1>c_2(k)\log^2 N)/\log^3N.$
While the estimate for $$p>1$$ is probably best possible (as was recently shown by Roth for $$p=2$$, the estimate for $$p=1$$ is probably not.
[This article was published in the book announced in this Zbl 0356.00004.]
Reviewer: W. M. Schmidt

##### MSC:
 11K38 Irregularities of distribution, discrepancy 11J71 Distribution modulo one 51M25 Length, area and volume in real or complex geometry
##### Citations:
Zbl 0356.00004; Zbl 0057.28604; Zbl 0274.10038