Roots of trigonometric polynomials and the Erd\H{o}s-Tur\'an theorem

Abstract

We prove, informally put, that it is not a coincidence that $\cos{(n \theta)} + 1 \geq 0$ 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 Erd\H{o}s-Tur\'an 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 $q(\theta) = \Re \sum_{k=0}^{n}{a_k e^{i k \theta}}$. If $q(\theta)$ has $\lesssim n^{\delta}$ roots for some $0 < \delta < 1/2$, then the roots of $p(z)$ do not frequently cluster in angle at scale $\sim n^{-(1-\delta)} \ll n^{-1/2}$.