Almost Sure Convergence of the Norm of Littlewood Polynomials

This paper concerns the L⁴ norm of Littlewood polynomials on the unit circle which are given by align*qₙ(z)=∑ₖ₌₀ⁿ⁻¹± zᵏ;align* i.e., they have random coefficients in \-1,1\ . Let align*||qₙ||₄⁴=1/2π∫₀2π|qₙ(eiθ)|⁴ dθ.align* We show that ||qₙ||₄/√n→ √ [4]2 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.