文摘
It is known that the discrepancy \({D_N\{kx\}}\) of the sequence \({\{kx\}}\) satisfies \({ND_N\{kx\} = O((\log N){(\log \log N)}^{1+\varepsilon})}\) a.e. for all \({\varepsilon > 0}\), but not for \({\varepsilon=0}\). For \({n_k=\theta^k}\), \({\theta > 1}\) we have \({ND_N\{n_kx\} \leqq (\Sigma_\theta +\varepsilon){(2N\log \log N)}^{1/2}}\) a.e. for some \({0 < \Sigma_\theta < \infty}\) and \({N\geqq N_0}\) if \({\varepsilon > 0}\), but not for \({\varepsilon < 0}\). In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed \({\Psi(N)}\) between \({(\log N){(\log \log N)}^{1+\varepsilon}}\) and \({{(N\log \log N)}^{1/2}}\) and for any \({\Sigma > 0}\), there exists a sequence \({\{n_k\}}\) of positive integers such that \({ND_N\{n_kx\} \leqq (\Sigma+\varepsilon)\Psi(N)}\) eventually holds a.e. for \({\varepsilon > 0}\), but not for \({\varepsilon < 0}\). We also consider a similar problem on the growth of trigonometric sums.