Littlewood polynomials with small [formula omitted] norm

Abstract

Littlewood asked how small the ratio ‖f‖4/‖f‖2 (where ‖⋅‖α denotes the Lα 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 7/64, 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 22/194.