An extremal property of Fekete polynomials

The Fekete polynomials are defined as

where $\left(\frac{\cdot}{q}\right)$ is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known $L_4$ norm out of the polynomials with $\pm 1$ coefficients.

The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity.

Theorem 0.1. Let $f(x)=a_1x+a_2x^2+\cdots +a_{N-1}x^{N-1}$ with odd $N$ and $a_n=\pm 1$. If

then $N$ must be an odd prime and $f(x)$ is $\pm F_q(x)$. Here $\omega:=e^{\frac{2\pi i}{N}}.$

This result also gives a partial answer to a problem of Harvey Cohn on character sums.