Let $f(\theta)$ be some $2\pi$-periodic function which takes the values $f(\theta) \in \{1,-1\}$. Further let $Q$ be some number which is rationally independent of $2\pi$ (More specifically take $Q/(2\pi)$ to be some Diophantine number).
I define a sequence $u_n \in \{1,-1\}$ with $n \in \mathbb{N}$ to be quasi-periodic if either:
- $u_n$ can be written as $u_n = f(Q n)$ subject to the above definitions.
- $u_n$ is obtained by taking the product of finitely many sequences $u_n = v_n w_n \ldots$ where $v_n, w_n, \ldots$ satisfy (1.), but are permitted to have different values of $Q$
It is my further supposition that if $u_n$ is quasiperiodic, it implies the sequence $v_n$ defined by $v_n = u_n v_{n-1}, v_1=1$ is also quasi-periodic.
I have tried various things playing around with the fourier structure of the series without particular success, so I won't report here--but I feel like this shouldn't be so hard to see as I am finding it.
Numerical experiments with taking $u_n = (2 s_n -1)$ where $s_n\in{0,1}$ is the Fibonacci word (a well-known quasi-periodic sequence, which satisfies (1.) with $Q/(2\pi) = (1+\sqrt{5})/2$), indicate than $v_n$ is quasi-periodic according to definition (1.) with $Q/(2\pi) \approx 0.690983$