Density of sequences of the form $x_n=f(n{)}^n$ in [0,1]

Abstract

In 2013, Strauch asked how various sequences of real numbers defined from trigonometric functions such as $x_n=(\cos n{)}^n$ distributed themselves$(\mathrm{mod}1)$. Strauch's inquiry is motivated by several such distribution results. For instance, Luca proved that the sequence $x_n=(\cos \alpha n{)}^n(\mathrm{mod}1)$ is dense in $[0,1]$ for any fixed real number $\alpha$ such that $\alpha /\pi$ is irrational. Here we generalise Luca's results to other sequences of the form $x_n=f(n{)}^n(\mathrm{mod}1)$. We also examine the size of the set $|\{n\le N:r<|\cos (n\pi \alpha ){|}^n\}|$ where $0<r<1$ and $\alpha$ are fixed such that $\alpha /\pi$ is irrational.