Littlewood Polynomials with Small $L^4$ Norm

Littlewood asked how small the ratio ||f || 4 /||f || 2 (where ||·|| α denotes the L α norm on the unit circle) can be for polynomials f having all coefficients in {1, −1}, as the degree tends to infinity. Since 1988, the least known asymptotic value of this ratio has been 4 7/6, which was conjectured to be minimum. We disprove this conjecture by showing that there is a sequence of such polynomials, derived from the Fekete polynomials, for which the limit of this ratio is less than 4 22/19.