An equidistributed sequence: $an^\sigma$ for $a\neq0$ and $\sigma$ noninteger

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.

You can find the answer in Theorem 3.5 of Chapter 1 in the textbook by Kuipers and Niederreiter (which is also cited by Problem 4.3 of Stein and Shakarchi's Book I).

For $1<\sigma < 2$, its exercise 2.23 of Chapter 1 also suggests to use the Van der Corput's trick (that is, Theorem 2.7 of Chapter 1). I record some details here.

Based on this, I am wonderering is there any induction proof without noting the structure of $f$ stated in Theorem 3.5.