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}} $?