{
  "id": "1903.09079",
  "version": "v1",
  "published": "2019-03-21T16:00:04.000Z",
  "updated": "2019-03-21T16:00:04.000Z",
  "title": "Roots of trigonometric polynomials and the Erdős-Turán theorem",
  "authors": [
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "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 $\\sim n^{-(1-\\delta)} \\ll n^{-1/2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-21T16:00:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "erdős-turán theorem",
      "real roots",
      "statement holds",
      "sign changes",
      "corresponding trigonometric polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}