Discrepancy with respect to weighted means of some sequences (Q1075365)

The discrepancy of a sequence \(\omega =(x_ n)\) of real numbers with respect to a positive weighted mean \(P=(p_ n)\) is defined by \[ D_ N(P;\omega)=\sup_{I}| (1/P(N))\sum^{N}_{n=1}\chi_ I(\{x_ n\})-| I| \quad |, \] where \(P(N)=\sum^{N}_{n=1}p_ n\), \(\{x_ n\}=x_ n-[x_ n]\) denotes the fractional part of \(x_ n\), and \(\chi_ I\) the characteristic function of the interval I with length \(| I|\). \textit{H. Niederreiter} and the reviewer [Manuscr. Math. 42, 85-99 (1983; Zbl 0498.10030)] and the reviewer [ibid. 44, 265-277 (1983; Zbl 0507.10040)] proved upper bounds for a class of sequences and means. These results are generalized. Furthermore the author proves for any \(\alpha\) of approximation type \(\eta\), \(P=(g'(n))\) \((g(t)\in C^ 2[1,\infty)\), g(t)\(\to \infty\), \(g'(t)\to const<1\) monotonically as \(t\to \infty)\) that for every \(\epsilon >0\) \[ D_ N(P;[g(n)]\alpha)=O((1/g(N))\int^{N}_{1}(g'(t))^ 2 dt)^{(1/\eta)- e}). \]