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 non-trivial 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. 



upper bound for an incomplete exponential sum
In general no non-trivial 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
$$
\sum\_{a=1}^{N}e(\frac{ak}{q}) = \frac{1-e(\frac{(N+1)a}{q})}{1-e(\frac{k}{q})}\ll\frac{q}{a},
$$
which is non-trivial for $a>q^\delta$.


