Ein metrischer Satz der C-Gleichverteilung (Q794701)

The authors establish an analog of the fundamental metric theorem of \textit{H. Davenport}, \textit{P. Erdős}, and \textit{W. J. LeVeque} [Mich. Math. J. 10, 311--314 (1963; Zbl 0119.28201)] in the context of continuously uniformly distributed (c.u.d.) functions mod 1. This leads to various interesting extensions of an earlier metric theorem of \textit{E. Hlawka} [Monatsh. Math. 74, 108--118 (1970; Zbl 0208.31304)]. For instance, if \(\alpha>0\) is given, then \(f(t^{\alpha})\) is a c.u.d. function mod 1 for almost all (in the sense of the Wiener measure) real-valued continuous functions \(f\) on \([0,\infty)\). Furthermore, \(\int^{t}_{0}f(u)\,du\) is a c.u.d. function mod 1 for almost all real-valued continuous functions \(f\) on \([0,\infty)\).