Let us define:
$$\varphi(x,n,t):=\frac{1}{n}\sum_{y=1}^n \left| \sum_{k=1}^n e^{2ik\pi (x-y)/n + 2i \sin(2k\pi/n) t} \right|$$
Does somebody have an idea how to prove that
$$ \sup_{x=1,...,n} \varphi(x,n,t) $$
remains bounded uniformly in $n$ and in $t$ (or at least for $t$ of order $n^2$) ?
This is obvious for $t=0$, since that quantity is uniformly equal to 1. Simulations in Scilab seem to confirm that the additional phase angle does not affect the boundedness. I would like to make computations rigorous.