Sets of block structure and discrepancy estimates (Q1388990)

Sequences \(\underline{x}= (x_n)_{n\in \mathbb{N}}\) on finite sets \(M\) and sequences \(\underline{f}= (f_n)_{n\in \mathbb{N}}\) of maps \(f_n:M\to M\) are considered. The author investigates the problem of which information about \(\underline{x}\) and \(\underline{f}\) is suitable for getting estimates for the discrepancy of the sequence \(\underline{f} (\underline{x})= (f_n(x_n))_{n\in\mathbb{N}}\). A qualitative answer to the above problem was given by the author in an earlier paper [\textit{R. Winkler}, Monatsh. Math. 126, 161-174 (1998)] by means of a characterization of uniform distribution preserving maps. In the present paper it is shown that a quantitative answer has to be more complicated and it cannot be given in terms of discrepancy alone. The author analyzes how a quantitative result can be given: It depends on so-called ``almost constant sequences'' in the sense of \textit{V. Losert} and \textit{H. Rindler} [Astérisque 61, 133-143 (1979; Zbl 0404.22004); see also \textit{H. Rindler}, Acta Arith. 35, 189-193 (1979; Zbl 0407.10041)], who extended a result of \textit{G. Rauzy} [Sémin. Théor. Nombres 1972/73, Univ. Bordeaux, Exp. No. 20 (1973; Zbl 0293.10018)]. For a detailed survey on related references see \textit{M. Drmota} and \textit{R. F. Tichy} [ Sequences, discrepancies and applications, Lect. Notes Math. 1651 (Springer 1997; Zbl 0877.11043)].