On the integer part function and uniform distribution mod 1 (Q1360710)

Let \(\lambda\geq 0\) and \(\xi\in\mathbb{R}\backslash \mathbb{Q}\). We say that \(\xi\) is \(\lambda\)-admissible if there is a \(C(\xi,\lambda)> 0\) such that \(q_{k+1}< C(\xi, \lambda)q^{1+\lambda}_k\) for any integer \(k\geq 0\), where \(q_k\) is the denominator of the \(k\)th convergent to \(\xi\). In the paper, the author obtains mainly the following two results on the upper estimate for discrepancy of some sequences: (i) Let \(1< c<3/2\), \(v= 4/(3- 2c)\) and \(0\leq \lambda\leq vc-3\). If \(\xi\) is \(\lambda\)-admissible and \(D_{c, \xi}(x)\) denotes the discrepancy of the sequence \(\xi[n^c]\) for \(n= 1,2,\dots,[x]\), then \(D_{c, \xi}(x)= O_{c, \xi}(x^{-1/v} \log x)\), where \([\alpha]\) denotes the integral part of \(\alpha\in\mathbb{R}\). (ii) For \(1<c< 7/6\), \(0<q<1\), let \(\Delta_{c, q}(x)\) denote the discrepancy of the sequence \([n^c](\log n)^q\) for \(n=1,2,\dots, [x]\). Then \(\Delta_{c, q}(x)= O_{c, q}((\log x)^{(q-1)/2})\). In the proof, the author uses results of \textit{P. Erdős} and \textit{P. Turán} [Proc. Akad. Wet. Amsterdam 51, 1146-1154 (1948; Zbl 0031.25402) and ibid. 1264-1269 (1948; Zbl 0032.01601)] on the discrepancy.