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

## TL;DR

This paper explores the relationship between the roots of trigonometric polynomials and the distribution of roots of algebraic polynomials, extending the Erd ext{"o}s-Turán theorem by linking root distribution to sign changes.

## Contribution

It establishes a novel connection between the rate of root equidistribution in angle and the number of sign changes in associated trigonometric polynomials.

## Key findings

- Roots of polynomials with few real roots are evenly distributed in angle.
- The distribution scale of roots is related to the number of sign changes in the corresponding trigonometric polynomial.
- Fewer sign changes imply less frequent clustering of roots in specific angular sectors.

## 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 $\sim n^{-(1-\delta)} \ll n^{-1/2}$.