Let $p$ be a prime. Let $\omega$ be a $p$-th root of unity. We know that $\chi_\alpha (x) = \omega^{\alpha\cdot x}$ are the additive characters of $\mathbb{Z}_p$.

I have a question about bounding incomplete sums of these characters.

Let $T$ be a subset of $\mathbb{Z}_p$ of size $(p-1)/2$. I want statistics on sums of the form: $$\sum_{t \in T} \chi_\alpha(t)$$

In the worst-case, we know that this sum can be as large as $c\cdot p$ where $c$ is some constant. (e.g. when $\alpha = 0$ or $\alpha =1$ and $T = \{ 1, 2, \dots, (p-1)/2 \}$ )

Can we say something stronger about the expectation? That is, for every subset $T$, the expectation: $$\mathbb{E}_{\alpha \gets \mathbb{Z}_p} \sum_{t \in T} \chi_\alpha(t)$$

For example, is it bounded by $p^{\frac{1}{2}}$?

The trivial estimate here would be $$\mathbb E\,\Bigg|\sum_{t\in T}\chi_\alpha(t)\Bigg| \le \Bigg( \mathbb E\,\Bigg| \sum_{t\in T} \chi_\alpha(t) \Bigg|^2 \Bigg )^{1/2} = \Bigg( \sum_{t_1,t_2\in T} \frac1p\sum_{\alpha\in\mathbb Z_p} \omega^{\alpha\cdot(t_1-t_2)} \Bigg)^{1/2} = \sqrt{|T|} \le p^{1/2}.$$ Does this answer your question?
• One can see that this bound is sharp to within a constant factor, as long as $|T| \leq p/2$, by considering a random subset $T$. Commented Oct 14, 2017 at 11:21
• Also, without the absolute values, one has ${\mathbb E} \sum_{t \in T} \chi_\alpha(t) = 1_{0 \in T}$. Commented Oct 14, 2017 at 16:36