Roots of trigonometric polynomials and the Erdős-Turán theorem

Abstract

We prove, informally put, that it is not a coincidence that $\cos (n\theta )+1\ge 0$ 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 $q(\theta )=\Re {\sum }_{k=0}^n{a}_k{e}^{ik\theta }$. 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 $\sim n^{-(1-\delta )}\ll n^{-1/2}$.