ALMOST SURE CONVERGENCE OF THE L⁴ NORM OF LITTLEWOOD POLYNOMIALS

This paper concerns the L⁴ norm of Littlewood polynomials on the unit circle which are given by n−1 qn(z) = X ±z^k; k=0 i.e., they have random coefficients in {−1, 1}. Let ||qn||⁴₄ = ₂¹_π^Z₀²π |qn(e^iθ)|⁴dθ. We show that ||qn||4/√n → ^√4 2 almost surely as n → ∞. This improves a result of Borwein and Lockhart in 2001 [7] who proved the corresponding convergence in probability. Computer-generated numerical evidence for the a.s. convergence has been provided by Robinson in 1997 [25]. 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.