Need to evalute a closed form expression of the following limit:
$$\lim_{n \to \infty} \frac{1}{n}\sum_{t = 1}^{n}e^{-k\cos^2(\omega t)}$$ where $k>0$ and $0<\omega<\pi$.
Empirically, I have observed that the limit exists and it does not depend on $\omega$, it depends only on $k$. Also, the value of the limit decreases as $k$ increases, which means the limit is a decreasing function in $k$. I need to prove that the limit is independent of $\omega$ and is a decreasing function in $k$. Please help me in finding the limit.
If $\omega$ is not a rational multiple of $\pi$, then, because $\omega j$ is equidistributed mod $\pi$, it can be shown that $$ \begin{align} \lim_{n\to\infty}\frac1n\sum_{j=1}^ne^{-k\cos^2(\omega j)} &=\frac1\pi\int_0^\pi e^{-k\cos^2(x)}\,\mathrm{d}x\tag{1a}\\ &=e^{-k/2}I_0(k/2)\tag{1b} \end{align} $$ where $I_0$ is the modified Bessel function of the first kind.
However, this does not holds when $\omega$ is a rational multiple of $\pi$. For example, $$ \hspace{-20pt}\lim_{n\to\infty}\frac1n\sum_{j=1}^ne^{-k\cos^2(\omega j)}=\left\{\begin{array}{} e^{-k}&\small\text{if $\omega$ is an integer multiple of $\pi$}\\ \frac{1+e^{-k}}2&\small\text{if $\omega$ is an odd multiple of $\frac\pi2$}\\ \frac{e^{-k}+2e^{-k/4}}3&\small\text{if $\omega$ is a multiple of $\frac\pi3$, but not of $\pi$}\\ \frac{1+e^{-k}+2e^{-k/2}}4&\small\text{if $\omega$ is a multiple of $\frac\pi4$, but not of $\frac\pi2$}\\ \frac{e^{-k}+2e^{-k\phi^2/4}+2e^{-k\phi^{-2}/4}}5&\small\text{if $\omega$ is a multiple of $\frac\pi5$, but not of $\pi$}\\ \frac{1+e^{-k}+2e^{-3k/4}+2e^{-k/4}}6&\small\text{if $\omega$ is a multiple of $\frac\pi6$, but not of $\frac\pi3$ or $\frac\pi2$} \end{array}\right. $$