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 $|\sum\_{k=1}^n \{ck^a\}-\frac{n}2|=O(n^\theta)$?



Distribution of $\{cn^a\}$
Exercise 2.23 in Kuipers and Niederreiter, Uniform Distribution of Sequences, goes: Use Theorem 2.7 to show that the sequence $\{\alpha n^{\sigma}\}$, $n=1,2,\dots$, $\alpha\ne0$, $1<\sigma<2$, is uniformly distributed modulo one. 


Theorem 2.7 is as follows. Let $a$ and $b$ be integers with $a<b$, and let $f$ be twice differentiable on $[a,b]$ with $f''(x)\ge\rho>0$ or $f''(x)\le-\rho<0$ for $x\in[a,b]$. Then $$\left|\sum\_{n=a}^be^{2\pi if(n)}\right|\le(|f'(b)-f'(a)|+2)\left({4\over\sqrt{\rho}}+3\right)$$


