I'm interested in linear combinations of cosines: $f(x) = \alpha_1 \cos(2\pi \theta_1 x) + \alpha_2 \cos(2\pi \theta_2 x) + \cdots + \alpha_k \cos(2\pi \theta_k x)\enspace,$ where $\alpha_i \in \mathbb{Z}$ and the $\theta_i$'s are linearly dependent irrational numbers (when they are rational numbers or irrational linearly independent, I have less trouble, as in the former case the function is periodic, and in the second, any value of each $\cos$ can be attained independently of the others).
When plotting functions like this, one sees that they are not periodic, but still, they are close to be. In particular, it seems that there are values $p$ and $\epsilon$ such that if $f(x) > \epsilon$, then $f(x+n\times p) > \epsilon$, at least for the first few $n$'s. I have two questions:
- Is this true?
- Is there a name for the pseudo-periodicity that $f$ seems to enjoy?
Thanks!