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I'm interested in seeing whether the following is true. Assume $u$ is uniform on $[0,1]$. For a fixed $x\in\mathbb{C}^n$ with $\|x\|_{2}=1$ we have \begin{align*} \mathbb{P}\Big\{\Big|\sum_{k=0}^{n-1}x_ke^{2\pi i u k^2}\Big|\ge (\log n)^\alpha\Big\}\le \frac{1}{n^\beta} \end{align*} for some $\alpha>0$ and $\beta>0$. Note that $x$ is a fixed vector. Larger $\beta$ and smaller $\alpha$ is of course better.

Please note that I do not want this result for all $x$. Just for a fixed x. That is we have a fixed x and then draw the random variable $u$.

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  • $\begingroup$ I don't think you should expect a result like this. You should probably expect that the $e^{2\pi i u k^2}$ behave like independently chosen points on the unit circle. Then the sum should have size something like $\sqrt n$? $\endgroup$
    – Anthony Quas
    Commented Sep 9, 2014 at 14:44
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    $\begingroup$ Just to add to the discussion. We know that if we had say $e^{2\pi i u_k}$ with $u_k$ i.i.d. on [0,1] the expression would be true. The question is does the chirp function create a similar effect. If instead of log a polynomial is required what is the smallest power. $\endgroup$
    – mohi
    Commented Sep 9, 2014 at 15:11
  • $\begingroup$ @ChristianRemling: Oops: I didn't notice that the $x$ was scaled to have $L^2$ norm of 1. $\endgroup$
    – Anthony Quas
    Commented Sep 9, 2014 at 18:19
  • $\begingroup$ Please also see a more recent question I posed in mathoverflow.net/questions/181302/… which is further evidence that the above maybe true $\endgroup$
    – mohi
    Commented Sep 19, 2014 at 17:17

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