Flat Littlewood polynomials exist

We show that there exist absolute constants $\delta > \delta > 0$ such that, for all $n \ge 2$, there exists a polynomial $P$ of degree $n$, with coefficients in $\{-1,1\}$, such that \[ \delta\sqrt{n} \le |P(z)| \le \Delta \sqrt{n} \] for all $z \in \mathbb{C}$ with $|z|=1$. This confirms a conjecture of Littlewood from 1966.