The irrationality measure of $\pi$ as seen through the eyes of $\cos (n)$

Abstract

For different values of $\gamma \ge 0$, analysis of the end behavior of the sequence $a_n=\cos (n{)}^{n^{\gamma }}$ yields a strong connection to the irrationality measure of $\pi$. We show that if $\mathrm{limsup}|\cos n{|}^{n^2}\ne 1$, then the irrationality measure of $\pi$ is exactly 2. We also give some numerical evidence to support the conjecture that $\mu (\pi )=2$, based on the appearance of some startling subsequences of $\cos (n{)}^n$.

13 pages, 6 figures