Diskrepanz von Ketten (Q1256506)

In this paper chains are considered; a chain is an element of the free module generated by the space \(E_s^{\mathbb N}\) of all sequences \(\omega(x(n))_{n\in\mathbb N}\) with elements \(x(n)\) in the \(s\)-dimensional unit interval \([0,1)^s\). In a recent paper \textit{E. Hlawka} [Colloq. Math. Soc. János Bolyai 13, 97--109 (1976; Zbl 0337.10037)] introduced a notion of discrepancy for such chains \(c= \sum_\omega c(\omega) \cdot \omega\): \[ D(N,c) = \sup_J \left\vert \sum_\omega c(\omega) \lambda(N,J,\omega) - \vert c\vert \vert J\vert \right\vert, \] where the supremum is taken over all \(s\)-dimensional subintervals \(J\) of \([0,1)^s\); \(\vert J\vert\) is the volume of \(J\) and \(\lambda(N,J,\omega) = (1/N) \sum_{k=1}^N \chi_J(x_\omega(k))\) and \(\vert c\vert = \sum_\omega c(\omega)\); \(\chi_J\) the characteristic function of \(J\). The main result of this paper is an estimation of this discrepancy like Roth's estimation for usual sequences, in detail it is proved \[ N \cdot D(N,c) \ge (1/n)\biggl\vert\bigl\vert c\bigr\vert\biggr\vert C_s (\log N)^{s /2} \] for infinitely many \(N\); \(C_s\) is a constant.