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 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.
[For the entire collection see Zbl 0356.00004.]