Assume you are given a length-$n$ vector $x\in\mathbb{C}$ with elements $x_0$ through $x_{n-1}$. Define the Fourier transform of $x$ as
$$ X(e^{j\theta}) = \sum_{k=0}^{n-1} x_k e^{-jk\theta}. $$
I'm interested in the maximum of its magnitude, i.e., on
$$ \max_{\theta} |X(e^{j\theta})|^2. $$
Are there any non-trivial results for general $x$?
Not sure if you find this trivial, but this is about the only thing you can say:
$$\left|X(e^{j\theta})\right|=\left|\sum_{k=0}^{n-1}x_ke^{-jk\theta}\right|\le\sum_{k=0}^{n-1}|x_k|$$