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Abstract

In 2013, Strauch asked how various sequences of real numbers defined from trigonometric functions such as $x_n=(\cos n)^n$ distributed themselves $\pmod 1$. Strauch’s inquiry is motivated by several such distribution results. For instance, Luca proved that the sequence $x_n=(\cos \alpha n)^n\pmod 1$ 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 $x_n=f(n)^n\pmod 1$. We also examine the size of the set $|\{n\leq N:r \lt |\!\cos n\pi \alpha |^n\}|$, where $0 \lt r \lt 1$ and $\alpha $ are fixed such that $\alpha /\pi $ is irrational.