Let $f=(f_0,\ldots,f_{n-1})$ be a vector in $V_n=\{\pm 1\}^n$. Let $F=(F_0,\ldots,F_{n-1})$ be its (discrete) Fourier transform defined by $$ F_k=\sum_{x=0}^{n-1} f_x \omega_n^{x k} $$ where $\omega_n=\exp(2 \pi i/n)$. Let $$\theta_n=\min \left( \max_{0\leq k\leq n-1}|F_k|: f \in V_n \right).$$ Is anything known about the growth rate of $\theta_n$ with $n$? I have quickly computed the values corresponding to $n=3,4,\ldots,7$ as $$ (n,\theta_n)=[(3,2/\sqrt{3}),(4,1),(5,3/\sqrt{5}),(6,\sqrt{2}),(7,3/\sqrt{7})]. $$ A lower bound on the growth rate would be fantastic, if it is known.

The question can also be stated as computing the minimal infinity norm of the Fourier transform of the following difference of indicator functions $$\chi(A)-\chi(A^c)$$ where $A^c$ is the complement of $A$, as the set $A$ ranges over all subsets of $[n]$ so has an arithmetic combinatorics flavour.