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}} $?



About Averages of Incomplete Additive Character Sums
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?


