Higher order oscillation and uniform distribution. (English) Zbl 1479.11134
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.
MSC:
11K65 | Arithmetic functions in probabilistic number theory |
37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
37A25 | Ergodicity, mixing, rates of mixing |