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

TL;DR

This paper investigates the distribution and density of sequences of the form $f(n)^n$ modulo 1, extending previous results on cosine-based sequences and analyzing the size of certain subsets related to these sequences.

Contribution

It generalizes prior results on cosine sequences to a broader class of functions $f(n)$, establishing density and distribution properties modulo 1.

Findings

Sequences of the form $f(n)^n$ are dense in [0,1] under certain conditions.

The size of sets where $| ext{cos}(neta)|^n$ exceeds a fixed threshold is characterized.

Generalization of Luca's results to a wider class of functions $f(n)$.

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 $|\{n\leq N:r<|\cos(n\pi\alpha)|^n\}|$ where $0<r<1$ and $\alpha$ are fixed such that $\alpha/\pi$ is irrational.