Littlewood Polynomials with Small $L^4$ Norm

Abstract: Littlewood asked how small the ratio $||f||4/||f||_2$ (where $||.||\alpha$ denotes the $L^\alpha$ 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 $\sqrt[4]{7/6}$, 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 $\sqrt[4]{22/19}$.