# On the distribution functions of two oscillating sequences

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

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 $Gx n$ 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 $x n {n \geq 1} = \{ \logn \cos\alpha n \} {n \geq 1}$ is uniformly distributed. Finally we calculate $Gx n$ in the case when $\frac{\alpha}{2 \pi} \in \mathbb{Q}$.

Keywords : uniform distribution oscillating sequences set of all distribution functions Distribution moduo 1

Author: Christoph Aistleitner - Markus Hofer - Manfred Madritsch -

Source: https://hal.archives-ouvertes.fr/