Littlewood Polynomials with Small L⁴ Norm

الأصل · EN

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 √[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.