On the distribution functions of two oscillating sequences

Abstract : We investigate the set of all distribution functions of two special sequences on the unit interval, which involve logarithmic and trigonometric terms. We completely characterise the set of all distribution functions $G(x_n)$ for $(x_n)_{n \geq 1} = (\{\cos (\alpha n)^n\})_{n \geq 1}$ and arbitrary $\alpha$, where $\{x\}$ denotes the fractional part of $x$. Furthermore we give a sufficient number-theoretic condition on $\alpha$ for which $(x_n)_{n \geq 1} = (\{ \log(n) \cos(\alpha n) \})_{n \geq 1}$ is uniformly distributed. Finally we calculate $G(x_n)$ in the case when $\frac{\alpha}{2 \pi} \in \mathbb{Q}$.
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Contributor : Manfred Madritsch <>
Submitted on : Monday, October 21, 2013 - 10:03:38 AM
Last modification on : Friday, October 12, 2018 - 2:29:50 PM
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Christoph Aistleitner, Markus Hofer, Manfred Madritsch. On the distribution functions of two oscillating sequences. Uniform Distribution Theory, Mathematical Institute of the Slovak Academy of Sciences, 2013, 8 (2), pp.157-169. ⟨https://math.boku.ac.at/udt/⟩. ⟨hal-00875091⟩

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