I want to prove that for al $p\geq1$ there exists $\alpha\in[0;\pi]$ such that the series $cos(2^n\alpha)$ is p-periodic.
The obvious answer is picking $\alpha=0$ so that the series is constant of value $1$ (or $\alpha=arccos(-\frac{1}{2})$). This is what is making me guess the question doesn't only want $p$ to be a period but the smallest period.
I believe it is impossible, as I've tried with various recipes ($\alpha=\frac{\pi}{2^p}$) to no success. I am however also unable to prove it is impossible.
I've also found out that $\alpha$ cannot be a multiple of $\pi$ over a power of 2, otherwise the series will eventually be constant.
The easiest way to make $\cos 2^n\alpha$ periodic is to make $2^n\alpha$ itself periodic modulo $2\pi$. For example, solving $2^p\alpha=\alpha+2\pi$ looks promisung ...