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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
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