Let $f,g :\Bbb{R}\to\Bbb{R}$ be two periodic functions with periods $T$ and $T'$.
What can be said, in general about the periodic behaviour of their sum $f+g$?
If $T/T'$ is rational then a common multiple of $T$ and $T'$ should be a period for $f+g$.
But what can we say about the smallest positive period of $f+g?$
And if, in the problem is too general,how about when $f,g$ are trigonometric functions of the form $\sin{(ax)}, \cos{(ax)}$, where $a$ is rational?
For example, what is the smallest positive period for $f(x)=\sin {(35x)}+\cos{(42x)}$? , It is $\frac{2\pi}7$, where $7$ is $\gcd(35,42)$, i think. But how do we prove it is the smallest?
Do you know some online materials specifically treating this? Especially on functions like $\sin{(mx)}+\sin{(nx)}$, $\sin{(mx)}+\cos{(nx)}$ where $m,n$ are integers, for the beginning.
