Irregularities of distribution. I
Several important new results in the theory of irregularities of point distributions are proved. Let \(\mathcal P=\{z_1,\dots,z_n\}\) be a distribution of \(n\) points in the \(K\)-dimensional unit cube \(U^K\), let \(A\) be a measurable set in \(U^K\) and define the discrepancy by \((\mu\) is the Lebesgue measure) \[ D[\mathcal P;A]=\sum_{x\in\mathcal P\cap A}1-n\mu(A). \] The main theorem says that for every \(\varepsilon >0\) there exists a ball \(A\) in \(U^K\) such that \[ | D[\mathcal P;A] | > c\cdot n^{1/2-1/2k-\varepsilon} \] with a constant \(c=c(K,\varepsilon)>0\). This result improves a well-known result of W. M. Schmidt (where the exponent \(K-1/2K(K+2)-\varepsilon\) was obtained instead of \(1/2-1/2k-\varepsilon\). For a detailed survey on irregularities of distributions we refer to the monographs of \textit{W. M. Schmidt} [Lectures on irregularities of distribution. Bombay: Tata Inst. Fundamental Research (1977; Zbl 0434.10031)] and of the author and \textit{W. W. L. Chen} [Irregularities of distributions. Cambridge etc.: Cambridge University Press (1987; Zbl 0617.10039)]. Furthermore the author proves lower bounds for the discrepancy with respect to convex bodies where rotations are allowed. This rotation discrepancy is always large and behaves like the square root of the surface area. Probabilistic arguments show that this lower bound is nearly best possible. The proofs of the theorems are based on an ingenious application of the Fourier transformation method. \{Part II, cf. Proc. Lond. Math. Soc., III. Ser. 56, No. 1, 1--50 (1988; Zbl 0644.10038)\}.
scientific article; zbMATH DE number 4025531