Akiyama, Shigeki; Jiang, Yunping Higher order oscillation and uniform distribution. (English) Zbl 1479.11134 Unif. Distrib. Theory 14, No. 1, 1-10 (2019). Summary: It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form \((e^{2\pi i\alpha\beta n g(\beta))_{n\in\mathbb N}}\), for a non-decreasing twice differentiable function \(g\) with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number \(\alpha\) and almost all real numbers \(\beta> 1\) (alternatively, for a fixed real number \(\beta > 1\) and almost all real numbers \(\alpha)\) and for all real polynomials \(Q(x)\), sequences \((\alpha\beta n g(\beta)+ Q(n))_{n\in\mathbb N}\) are uniformly distributed modulo 1. Cited in 3 Documents MSC: 11K65 Arithmetic functions in probabilistic number theory 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37A25 Ergodicity, mixing, rates of mixing Keywords:higher order oscillating sequence; uniformly distributed modulo 1 Citations:Zbl 1439.11191; Zbl 1426.37008 PDFBibTeX XMLCite \textit{S. Akiyama} and \textit{Y. Jiang}, Unif. Distrib. Theory 14, No. 1, 1--10 (2019; Zbl 1479.11134) Full Text: DOI arXiv