There was a question recently about if $\left|\sum_{k=0}^n \sin k\right| \leq M$ for some $M$.
I wonder, how far could it be generalized? My idea is:
Let $f$ be continuous(*), bounded, periodic function with an irrational period $s$ such that $\int_0^s f(x)\,\mathrm{d}x = 0$. Then there is $M$ such that $$\left|\sum_{k=0}^n f(k)\right| \leq M.$$
(*) Or rather uniformly or absolutely continuous?
The informal idea is that if the period is irrational, the terms of the sum will be "randomly" placed in the interval and since $f$ is continuous and its integral is 0, they should eventually cancel out, just as if we integrated $f$ using Monte-Carlo integration.
If it's true, how to prove such a statement? If not, why, and how do we need to strenghten the premisses?
