Let $n_1 < \dots < n_N$ be positive integers. Assume we don't know anything about their actual values. What is the best general upper bound we can give for $$ \mu \left( x \in [0,1] : ~\left|\sum_{k=1}^N e^{2 \pi i n_k x} \right| > \kappa \sqrt{N} \right)? $$ Here $\mu$ is Lebesgue measure on $[0,1]$, and $\kappa$ is a parameter (imagine $\kappa$ to be somewhere around $\sqrt{\log N}$).

Of course we can use $\|\sum_{k=1}^N e^{2 \pi i n_k x}\|_2 = \sqrt{N}$ plus Chebyshev's inequality, to get $\mu \leq \kappa^{-2}$. But is there anything smarter we can do? Taking other norms than $L^2$-norm does not work, since in general these norms can get very large and the result will be worse than using $L^2$ norm. For example when $n_1 , \dots, n_N = 1, \dots, N$, we have an $L^4$ norm as large as $N^{3/4}$ - however, the actual size of the exceptional set is very small, so arguing with $L^4$ norms would be stupid in that situation. So is there any general estimate which beats the $L^2$ estimate?