In the sum
$$\sum_{n=1}^N \left(\{n \alpha\}-\frac{1}{2}\right)$$
where $\{x\}$ indicates the fractional part of $x$ and $\alpha$ is an irrational number, Koksma inequality suggests an order of $\log^2 N $ if we use a weak bound for the discrepancy (Kuipers and Niederreiter, Chapter 2, "Special sequences" section) and a stronger one using the bound of $\log N (\log \log N)^{1+\varepsilon}$ that is valid for almost all $ \alpha$ (Khintchine). Some details on this are reported in this previous question.
I wonder what can we say about the order of higher-degree sums, as
$$\sum_{n=1}^N\left( \{n \alpha\}^2-\frac{1}{3}\right)$$ $$\sum_{n=1}^N \left(\{n \alpha\}^3-\frac{1}{4}\right)$$
generalizing to any exponent $k$. Should we use Erdös-Turan inequality to estimate the discrepancy?
I am not searching particularly strong bounds, but only a way to get a formal proof for even weak O terms. After estensive search, I did not find anything on these higher-degree sequences in the literature.
