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$L^4$ norm of Littlewood polynomials
This paper concerns the 
$L^4$ norm of Littlewood polynomials on the unit circle which are given by i.e., they have random coefficients in 
$$ \begin{align*}q_n(z)=\sum_{k=0}^{n-1}\pm z^k;\end{align*} $$
$\{-1,1\}$. Let We show that 
$$ \begin{align*}||q_n||_4^4=\frac{1}{2\pi}\int_0^{2\pi}|q_n(e^{i\theta})|^4 d\theta.\end{align*} $$
$||q_n||_4/\sqrt {n}\rightarrow \sqrt [4]{2}$ 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.
A Littlewood polynomial is a polynomial in 
$\mathbb{C}\left[ z \right]$ having all of its coefficients in 
$\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small 
${{L}^{q}}$ normon the complex unit circle. We consider the Fekete polynomials
,
$${{f}_{p}}(z)=\sum\limits_{j=1}^{p-1}{(j|p){{z}^{j}}},$$
where 
$p$ is an odd prime and 
$(.|p)$ is the Legendre symbol (so that 
${{z}^{-1}}{{f}_{p}}(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of ${{L}^{q}}$ and 
${{L}^{2}}$ norm of 
${{f}_{p}}$ when 
$q$ is an even positive integer and 
$p\to \infty $. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the 
${{L}^{4}}$ norm of these polynomials.
