Littlewood polynomial

Summary

A Littlewood polynomial is a polynomial all of whose coefficients are $+1$ or $-1$. On the unit circle such a polynomial of degree $n$ has mean square $n + 1$, so its typical size there is $\sqrt{n+1}$. Littlewood asked how close to constant its absolute value on the circle can be. This question about flat polynomials links harmonic analysis, combinatorics and signal design. The Rudin–Shapiro polynomials are a classical family that is flat from above.