Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) we have $|\sum_{k=1}^n \{cn^a\}-\frac{n}2|=O(n^\theta)$?

Assume that $1<a<2$ and $c\ne 0$ is a real number. What is known about the distribution of the sequence $cn^a$ modulo 1? Say, is it true that for certain $\theta<1$ (depending on $a$ and $c$) we have $|\sum_{k=1}^n \{ck^a\}-\frac{n}2|=O(n^\theta)$?