Let $f : \mathbb R\to\mathbb C$ be an $1$-periodic and sufficiently smooth function, which has zero average, and let $\alpha$ irrational. We know the following:

a. $\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n f(k\alpha)=0$.

b. The sequence $s_n=\sum_{k=1}^n f(k\alpha)$ is not necessarily bounded for every such $\alpha$. In fact, we are (to the best of my knowledge) certain that $s_n$ is bounded for every such $\alpha$ only if $f$ a trigonometric polynomial.

c. Combining a. & b. we obtain that $s_n=o(n)$.

Is it the best estimate we can have for $s_n$?

This is a follow-up question to my earlier question:

Does equidistribution of zero average, due to irrationality, imply boundedness?