{
  "id": "2204.13541",
  "version": "v1",
  "published": "2022-04-28T14:44:27.000Z",
  "updated": "2022-04-28T14:44:27.000Z",
  "title": "On the Balog-Ruzsa Theorem in short intervals",
  "authors": [
    "Yu-Chen Sun"
  ],
  "categories": [
    "math.NT",
    "math.CA",
    "math.CO"
  ],
  "abstract": "In this paper we give a short interval version of the Balog-Ruzsa theorem concerning bounds for the $L_1$ norm of the exponential sum over $r$-free numbers. As an application, we give a lower bound for the $L_1$ norm of the exponential sum defined with the M\\\"obius function. Namely we show that $$\\int_{{\\mathbb T}} \\left|\\sum_{|n-N|<H} \\mu(n)e(n \\alpha)\\right| d \\alpha \\gg H^{\\frac{1}{6}}$$ when $H \\gg N^{\\frac{9}{17} + \\varepsilon}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-28T14:44:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exponential sum",
      "short interval version",
      "balog-ruzsa theorem concerning bounds",
      "free numbers",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}