Let $\mathbf{x}\in\mathbb{C}^n$ with $\|\mathbf{x}\|=1$ with $n<\frac{N}{2}$. I am interested in a bound of the form \begin{equation*} \mathbb{E}\Big\{\max_{k\in\{1,2,\ldots,n\}}\Big|\sum_{a=1}^ne^{-2\pi i\big(\frac{(a-1)(k-1)}{N}+\gamma(a-1)^2\big)}x_a\Big|^2\Big\}\le ?, \end{equation*} where $\gamma$ is a uniform random variable in the interval $[0,1]$. In particular I'm interested in getting a constant or log factor for the question mark. Please note that for a fixed k this is true but I want this for the maximum over all k. Also I don't want this to be true for all $\mathbf{x}$ rather for a fix $\mathbf{x}$. That is, given an $\mathbf{x}$ I draw an indepenedent random variable $\gamma$ uniform in the interval $[0,1]$.
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