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 $\sqrt[4]{22/19}$.

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