Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$ denotes the Fourier transform of $1_A$, then what can be said about $\mathbb{E}\left( \sup_{t\neq 0} |\hat{1}_A(t)|\right)?$ Do we have an upper bound of the form $O\left(\frac{\log p}{\sqrt{p}}\right)$ as $p$ goes to infinity?
Alternate wording: For each subset $A$ of the $p$ roots, let $Sum(A)$ to denote the sum of the elements in $A$, and look at $A^t:=\{a^t:\ a\in A\}$ for integers $0
Remarks: This question originated from an optional homework problem in my Arithmetic Combinatorics class. I tried some fairly long and drawn out things that did not work. Also, note that since $p$ is prime, taking the power for $0