I am looking for a hint on a problem in Stein and Shakarchi's book on fourier analysis. The problem asks for a proof that the fractional part of the sequence $an^\sigma$ is equidistributed for any $\sigma>0$ not an integer.
There is a much easier exercise in the book which only asks for a proof when $0 < \sigma < 1$, which has been asked elsewhere on the site, and I have also solved.
Currently, I don't have much progress to show, as most of what I have tried doesn't work. I've been trying to use Weyl's criterion to make progress.
In particular, for $\sigma < 1$ we can learn a lot from comparing it to the integral $\int_1^N{e^{2\pi ibx^\sigma}dx}$. Intuitively, I think thisis because the period is increasing, and sampling at the integers provides an increasingly accurate estimation of the integral over $[k, k+1]$. For $\sigma >1$, this approach seems to fail: as the period decreases, at least over $[k,k+1]$ sampling at an individual point tells you less.
Thank you :)
P.S. this is not a homework problem. I am studying the book in addition to my current maths undergraduate degree