Roots of trigonometric polynomials and the Erdős-Turán theorem

Abstract

We prove, informally put, that it is not a coincidence that $\cos{(n θ)} + 1 \geq 0$ and that the roots of $z^n + 1 =0$ are uniformly distributed in angle -- a version of the statement holds for all trigonometric polynomials with `few' real roots. The Erdős-Turán theorem states that if $p(z) =\sum_{k=0}^{n}{a_k z^k}$ is suitably normalized and not too large for $|z|=1$, then its roots are clustered around $|z| = 1$ and equidistribute in angle at scale $\sim n^{-1/2}$. We establish a connection between the rate of equidistribution of roots in angle and the number of sign changes of the corresponding trigonometric polynomial $q(θ) = \Re \sum_{k=0}^{n}{a_k e^{i k θ}}$. If $q(θ)$ has $\lesssim n^δ$ roots for some $0 < δ< 1/2$, then the roots of $p(z)$ do not frequently cluster in angle at scale $\sim n^{-(1-δ)} \ll n^{-1/2}$.