Almost sure convergence of the $L^4$ norm of Littlewood polynomials

• Published in: Canadian mathematical bulletin 2024-09, Vol.67 (3), p.872-885
• Main Authors: Duan, Yongjiang, Fang, Xiang, Zhan, Na
• Format: Article
• Language: English
• ISSN: 0008-4395; 1496-4287
• DOI: 10.4153/S0008439524000213

This paper concerns the $L^4$ norm of Littlewood polynomials on the unit circle which are given by $$ \begin{align*}q_n(z)=\sum_{k=0}^{n-1}\pm z^k;\end{align*} $$ i.e., they have random coefficients in $\{-1,1\}$ . Let $$ \begin{align*}||q_n||_4^4=\frac{1}{2\pi}\int_0^{2\pi}|q_n(e^{i\theta})|^4 d\theta.\end{align*} $$ We show that $||q_n||_4/\sqrt {n}\rightarrow \sqrt [4]{2}$ almost surely as $n\to \infty $ . 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.