Let $q$ be a large prime and $\delta\in(0,1)$. Let $k$ be an integer which is not a multiple of $q$. Define $e(x)=e^{2\pi ix}$. Can we get any nontrivial upper bounds for $$\sum_{a=1}^{q^{1\delta}}e(\frac{k}{q}a) $$? A bound like $q^{1\delta\epsilon}$ would be good enough. I have searched literature in Weyl's sum and did not find any good bounds.

5$\begingroup$ As written, this is just a finite geometric series, which has an elementary closed form that can be used to estimate the size of the sum. Is that really the sum you mean? $\endgroup$ – Noam D. Elkies Mar 7 '14 at 23:32

$\begingroup$ Yes. I have tried the geometric series formula and was unable to use that form to get a nontrivial upper bound in terms of a power of q. Maybe I need to take the magnitude of k into consideration. $\endgroup$ – Tony B Mar 8 '14 at 21:43
In general no nontrivial bound exists. Suppose e.g. that $a=1$. Then for all $k$ in the range of summation we have $\frac{ka}{q}\in[0, q^{\delta}]$, thus $e(\frac{ka}{q})=1+\mathcal{O}(q^{\delta})$.
If you have more information on $a$ you might use $$ \sum_{a=1}^{N}e(\frac{ak}{q}) = \frac{1e(\frac{(N+1)a}{q})}{1e(\frac{k}{q})}\ll\frac{q}{a}, $$ which is nontrivial for $a>q^\delta$.

$\begingroup$ Thank you for your answer. It seems that you messed up with a and k somewhere. $\endgroup$ – Tony B Mar 14 '14 at 23:04
