Question: I want to bound
\[ \sum _{q\leq \gamma }\int _{\pm 1/q\gamma }\left |\int _{x}^{2x}e(u\beta )u^{iT}du\right |^2d\beta \] or maybe just \[ \int _{\pm \delta }\left |\int _x^{2x}e(u\beta )u^{iT}du\right |^2d\beta .\]
Here $x,\gamma ,T>0$ are large with $\gamma ,T$ powers of $x$ less than 1, and $e(z)=e^{2\pi iz}$. What bound is possible?
A simpler case: We have \[ \int _{\pm \delta }\left |\int _x^{2x}e(u\beta )du\right |^2d\beta \leq x^2\int _{\pm 1/x}d\beta +2\int _{1/x}^\infty \frac {d\beta }{\beta ^2}\ll x\] More precise question: In view of the simpler case, what I would like to know is whether I can use the $T$ oscillation to get \[ \int _{\pm \delta }\left |\int _x^{2x}e(u\beta )u^{iT}du\right |^2d\beta \ll \text { smaller than }x.\] Perhaps it is $\ll x/\sqrt T$ or even $\ll x/T$?
The exact relation between $x,T,1/\delta $ seems important, and I think I want to be able to take something like $T\approx x^{1/3}$, $\delta \approx 1/x^{2/3}$. (And ideally that the whole quantity (with sum) is $\ll x^{2/3}$). But even if the sizes aren't quite this I'd be happy to know something.
Some comments: Since \[ e(u\beta )u^{iT}=e\left (f(u)\right )\] with \[ f(u)\approx u\beta +T\log u\] \[ f'(u)\approx \beta +T/u\] \[ f''(u)\approx T/u^2\] I hoped to use some of the basic results from the theory of oscillatory integrals, e.g. \[ \int _x^{2x}e(u\beta )u^{iT}du\ll x/\sqrt T\] or for $\beta $ away from $T/x$ it is even $\ll x/T$. I have tried to integrate first over $\beta $ and then to use these results but I still can't get anywhere.