Prove that $\{an^\sigma\}$ is equidistributed in $ [0,1) $,if $\sigma>0$ is noninteger and $a\neq 0$.
I know how to solve this problem if $\sigma <1$ , so it is not a duplicate of Equidistribution of $an^\sigma$.
But the ideas used there , fail to solve the general case. For solving this, I need to bound $\int ^ {n+1}_n |e^{2\pi ib n^{\sigma}}-e^{2\pi ib x^{\sigma}}|$ with a tight enough bound.
Is there any hint? Thanks.