On the distribution functions of two oscillating sequences

Christoph Aistleitner (1) , Markus Hofer (1) , Manfred Madritsch (2)

1 Institut fur Mathematik
2 IECL - Institut Élie Cartan de Lorraine

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 $\frac{\alpha}{2 \pi} \in \mathbb{Q}$.

Mots clés: Distribution moduo 1; set of all distribution functions; uniform distribution; oscillating sequences

Domaines: Théorie des nombres [math.NT]

Christoph Aistleitner, Markus Hofer, Manfred Madritsch. On the distribution functions of two oscillating sequences. Uniform Distribution Theory, 2013, 8 (2), pp.157-169. ⟨hal-00875091⟩