Almost Sure Convergence of the Norm of Littlewood Polynomials

Abstract This paper concerns the L⁴ norm of Littlewood polynomials on the unit circle which are given by align*qₙ (z) =₊=₀^n-1 zᵏ;align* i. e. , they have random coefficients in \-1, 1\. Let align*||qₙ||₄⁴=12₀^2|qₙ (e^i) |⁴ d. align* We show that ||qₙ||₄/ n 42 almost surely as n. This improves a result of Borwein and Lockhart (2001, Proceedings of the American Mathematical Society 129, 1463–1472), who proved the corresponding convergence in probability. Computer-generated numerical evidence for the a. s. convergence has been provided by Robinson (1997, Polynomials with plus or minus one coefficients: growth properties on the unit circle, M. Sc. thesis, Simon Fraser University). We indeed present two proofs of the main result. The second proof extends to cases where we only need to assume a fourth moment condition.