Almost sure convergence of the $L^4$ norm of Littlewood polynomials
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Bibliographic record
Abstract
Abstract 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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
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| Insufficient payload (model declined to judge) | 0.011 | 0.001 |
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