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

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