## Abstract

This paper concerns the L^{4} 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||^{4}_{4} = _{2}^{1}_{π}^{Z}_{0}^{2}π |qn(e^{iθ})|^{4}dθ. 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.

Original language | English |
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Journal | Canadian Mathematical Bulletin |

DOIs | |

State | Accepted/In press - 2024 |

## Keywords

- almost sure convergence
- L norm
- Littlewood polynomial
- Serfling’s maximal inequality

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