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