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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. Then, what is the expectation of the maximum of the sum over $t$? How well can we bound $$\mathbb{E}_A\left(\sup_{t} \ \left|Sum\left(A^t\right)\right|\right).$$

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 corresponds exactly to the automorphisms.

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    This might be a silly question, but how do you take the sup of a set of complex numbers ?2012-01-19
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    Just an aside regarding terminology: It might be good to be cautious with the use of the terms "characteristic function" and "Fourier transform" here. Indeed, in probability the use of the term "[characteristic function](http://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29)" is interestingly enough a Fourier transform and, the term "[indicator function](http://en.wikipedia.org/wiki/Indicator_function)" is used for functions like $1_A$ that are usually referred to as "characteristic functions" in other branches.2012-01-20
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    @JoelCohen: Sorry! My mistake, I forgot some critical absolute value bars.2012-01-20
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    Disregard my comment last night, I was thinking of a similar but different problem.2012-01-20

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