$L^q$ norms of Fekete and related polynomials

TL;DR

This paper investigates the asymptotic behavior of the $L^q$ norms of Fekete polynomials and related Littlewood polynomials, providing explicit formulas for their limits as the degree grows, which advances understanding of their approximation properties.

Contribution

It derives explicit recursive formulas for the limits of $L^q$ to $L^2$ norms of Fekete and related polynomials for infinitely many $q$, a novel result in the field.

Findings

Explicit formulas for the limits of $L^q$/$L^2$ norms as degree tends to infinity.

Generalization of previous results on the $L^4$ norm to all even positive integers $q$.

Extension to polynomials from additive characters of finite fields.

Abstract

A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erd\H{o}s, 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 $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\mid p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\mid p)$ 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 $f_p(z)$ 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.