Let $p$ be a large prime, and $S$ a set of non-negative integers less than $p$. Consider the distribution $D_{S,p}$ over complex numbers defined as $\sum_{x\in S}e^{i2\pi x y/p}$ for $y$ chosen uniformly at random from the set of non-negative integers less than $p$.
Is it possible to show, for arbitrary sets $S$, that $D_{S,p}$ is almost always very close to $0$? For example, it seems plausible that there are positive constants $B,C$ such that, regardless of $S$, $\Pr[|D_{S,p}|\geq \lambda \sqrt{p}]\leq B e^{-C \lambda^2}+B/p$.
Note that in the worst case, $D_{S,p}$ can be as large as $\Omega(p)$, say when $S$ is a set of size $\approx p/2$ and we happen to chose $y=0$. However, the randomization by $y$ seems to make these large cases very rare. If we were to treat each term in the sum as independent random variables, the hypothesized inequality above would follow from known tail-bounds for sums of independent random variables. However, these terms are quite correlated, so it's unclear how to give non-trivial tail bounds for this sum.